5.3: The Fundamental Theorem of Calculus EX. Given the graph of f, sketch the graph of x 0 f(t) dt. The definite integral gives the area under the curve. EX 2. Find the derivative of g(x) = x 0 + t 2 dt. Simplest use of FTC: derivative of integral is original function. EX 3. A real-life function defined as an integral. Stewart wants us to believe that integration is important. EX 4. Find d dx x 4 sec t dt. How to apply FTC when the upper limit of integration is more complicated than just x. EX 5. Evaluate the integral 3 e x dx. Simplest use of FTC2: evaluating definite integrals. EX 6. Find the area under the parabola y = x 2 from 0 to. Next simplest use of FTC2: finding areas under curves. You should learn some antiderivatives by heart. (See table on page 392.) EX 7,8. (Nothing new.) 3 3 x EX 9. What s wrong with dx = = x2 3 = 4 3? It s important that the integrand be (somewhat) continuous. If the sign of the answer is predictable, it s a good sanity check. Math 0 Lab ET3 2009 January 4
2 5.5: The Substitution Rule EX. Find x 3 cos(x 4 + 2) dx. Substitute for one part of the integrand when its derivative is also present (and can be put with the dx). When looking for the derivative of one part, ignore the presence/absence of constant multiples. Substitute for the inner function in a composition. For indefinite integrals, return to the original variable! (Very common error.) 2x EX 2. Find + dx. Linear substitution is cheap. (The derivative is always there, because we don t care about constant multiples.) EX 3 6. (Nothing new.) EX 7. Evaluate 4 0 2x + dx. For definite integrals, change limits when substituting! (Very common error.) EX 8,9. (Nothing new.) EX 0. Evaluate 2 2 (x 6 + ) dx using the evenness of the integrand. Using evenness doesn t change the calculation much. EX. Evaluate tan x + x 2 dx using the oddness of the integrand. + x4 Using oddness makes this integral easy; doing it directly (i.e., by finding an indefinite integral and using FTC2) is hugely more difficult, maybe even impossible. Now and then you ll meet a definite integral which can only be evaluated with some trick based on the specialness of the limits of integration, but it s pretty rare. 2009 January 4 Math 0 Lab ET3
3 6.: Areas Between Curves EX. Find the area of the region bounded above by y = e x, bounded below by y = x, and bounded on the sides by x = 0 and x =. Finding areas between curves (well, just graphs of functions for now) using a definite integral. EX 2. Find the area of the region enclosed by the parabolas y = x 2 and y = 2x x 2. The limits of integration might not be given explicitly. EX 3. Find the approximate area of the region bounded by the curves y = x/ x 2 + and y = x 4 x. Apparently the lesson is if you don t know, guess. Not a good lesson. EX 4. What does area on a graph of velocity mean? EX 5. Find the area of the region bounded by the curves y = sin x, y = cos x, x = 0, and x = π/2. What to do if you really want area and not signed area. 6.2: Volumes EX. The volume of a sphere. Computation of volume by taking cross-sections. Choose a coordinate system respecting the symmetry of the problem. You might have known this formula already; now you know why it s true. EX 2. Find the volume of the solid obtained by rotating about the x-axis the region under the curve y = x from 0 to. Generalization of example to solids of revolution. (Disks method.) EX 3. Find the volume of the solid obtained by rotating the region bounded by y = x 3, y = 8, and x = 0 about the y-axis. Take cross-sections perpendicular to the axis of revolution and integrate along that axis. (Disks/washers method only! Shells method is vice versa; see 6.3 below.) Math 0 Lab ET3 2009 January 2
4 EX 4. Find the volume of the solid obtained by rotating the region bounded by y = x, y = x 2 about the x-axis. When the region being rotated doesn t extend to the axis of revolution. (Washers.) EX 5,6. Find the volume of the solid obtained by rotating the region in example 4 about the line y = 2. Again, for the line x =. The radii involved are the distance to the axis of revolution, wherever that axis is. Compute them accordingly. EX 7 9. Find the volume of various solids (not solids of revolution). Cross-sections may have other shapes, but the same overall plan applies: cut up the solid into slices and integrate the area of the slices. You will have to improvise some details of the computation. You get to choose the coordinate system and in what direction to slice up the solid. Choose wisely. Two rules of thumb: if there s a circle, put the origin at its centre; if there s a cone or pyramid or some such thing, put the origin at its vertex, take cross-sections perpendicular to the axis, and be prepared to use similar triangles. 6.3: Volumes by Cylindrical Shells EX. Find the volume of the solid obtained by rotating the region bounded by y = 2x 2 x 3 and y = 0 about the y-axis. Cut up the region with slices parallel to the axis of revolution and integrate perpendicular to it. (Contrast 6.2, ex. 3 above.) In this problem, slicing parallel to the axis of revolution yields slices whose heights are easy to compute, while slicing perpendicular (as in the washers method) yields slices whose heights are hard to compute. In other problems the reverse happens. Be alert. EX 2 4. (Nothing new.) 2009 January 2 Math 0 Lab ET3
5 7.: Integration by Parts EX. Find x sin x dx. Integration by parts turns one integral into another, hopefully an easier one. There might be many choices of u and dv. Choose wisely. Try this one with u = sin x and dv = x dx, to see that a bad choice makes the integral more complicated. List the functions whose derivatives and integrals you know. Which ones get simpler when differentiated? When integrated? Which ones stay the same? EX 2. Evaluate ln x dx. u can be the whole integrand! This is a good trick when you know the derivative but need the integral. EX 3. Find t 2 e t dt. Sometimes you have to integrate by parts more than once. Make the same choice of u and dv each time. Making the opposite choice the second time just gets you back where you started. EX 4. Evaluate e x sin x dx. Neither part gets simpler when differentiated or when integrated; it looks hopeless. If you get the original integral back with some weird factor, you can solve for it. EX 5. Calculate 0 tan x dx. The same trick as in example 2. Shows how to handle the limits in a definite integral. EX 6. Prove the reduction formula sin n x dx = n cos x sinn x + n sin n 2 x. n The same trick as in example 4. As Stewart says, this formula can be used several times to reduce the power of sin until the integral is easy; but see the next section for other approaches. Math 0 Lab ET3 2009 January 28
6 7.2: Trigonometric Integrals EX,2. Evaluate cos 3 x dx and sin 5 x cos 2 x dx. A very common way to substitute: put one cos x with the dx, convert the rest to sin x using Pythagoras, then substitute u = sin x. This works when there s an odd amount of cos. The same with u = cos x, when there s an odd amount of sin. π EX 3,4. Evaluate sin 2 x dx and sin 4 x dx. 0 The half-angle identities can save the day when there s even amounts of both sin and cos. Know your trig identities! The strategy boxes on pages 462 3 are a little overwrought. Just do enough of the substitution in your head to see whether it ll work out. EX 5,6. Evaluate tan 6 x sec 4 x dx and tan 5 θ sec 7 θ dθ. The same idea as examples and 2, but for integrals in tan and sec. Stewart s evaluation of sec x dx on page 464 is highly weird; nobody would think to do it that way. I ll show you a more natural way later. EX 7. Find tan 3 x dx. Stewart s solution is fine, but there are other ways. Try writing tan x = sin x cos x and using the plan of examples and 2. (There s often more than one way; see exercise 56 in this section.) EX 8. Find sec 3 x dx. This one is right at the limit of what these techniques can do. Look how many ideas from previous examples conspire to make up this solution. EX 9. Evaluate sin 4x cos 5x dx. Another small family of trig integrals which become simple with the right identity. Know your trig identities! 2009 January 28 Math 0 Lab ET3
7 7.3: Trigonometric Substitution 9 x 2 EX. Evaluate dx. x 2 Stewart does two things at once here. Try them separately, thus: First, force-factor out the 9, i.e., write 9 x 2 = 3 ( x 3 )2, and then substitute for x 3. Then you re looking at u 2 (with some other stuff); substitute u = sin θ. Making a trig substitution yields a trig integral; use techniques from 7.2. EX 2. Find the area enclosed by the ellipse x2 a 2 + y2 b 2 =. Using lots of stuff from earlier sections here: how to compute areas, how to substitute in definite integrals, tricks for trig integrals. Handle parameters like a and b just like they were concrete numbers. EX 3. Find x 2 x 2 + 4 dx. Same as example, but using tan 2 θ + = sec 2 θ and substituting u = tan θ. x EX 4. Find x 2 + 4 dx. Don t forget about simpler substitutions. dx EX 5. Evaluate x 2 a. 2 Same as usual, but now using the identity sec 2 θ = tan 2 θ. (Hyperbolic substitution is nifty and useful, but most instructors skip it.) EX 6. Find 3 3/2 0 x 3 (4x 2 dx. + 9) 3/2 Recognize square roots in disguise. Change the limits when substituting in a definite integral. x EX 7. Evaluate dx. 3 2x x 2 How to get rid of the middle term in a quadratic. Math 0 Lab ET3 2009 February 4
8 7.4: Integration of Rational Functions by Partial Fractions EX 8. The method of partial fractions goes like this: If the numerator has big degree (as big as or higher than the denominator), divide. Factor the denominator. Write out the partial fraction decomposition: Factors like (x ) yield terms like Factors like (x ) 3 yield terms like A x. A x + B (x ) 2 + C (x ) 3. Factors like (x 2 +x+) yield terms like Ax + B x 2 (use the discriminant to check + x + for irreducibility!). Factors like (x 2 + x + ) 2 Ax + B yield terms like x 2 + x + + Cx + D (x 2 + x + ) 2. Solve for the coefficients: clear fractions, multiply everything out, identify coefficients of like powers, solve the resulting system of equations. (Or: multiply everything out, then plug in special values.) (Check that your coefficients are right by adding it all back up again.) Integrate the individual terms: A Terms like : yield logs. x A Terms like x 2 : substitute with tangent. + Ax Terms like x 2 : substitute for the denominator. + Terms like Ax + B x 2 : split them in two. + Ax + B Terms like x 2 : complete the square and make the linear substitution. + x + x + 4 EX 9. Evaluate dx x Sometimes substituting for a bad thing will make it go away and stop bothering you. 2009 February 4 Math 0 Lab ET3
9 7.8: Improper Integrals 0 EX,2. Determine whether dx is convergent or divergent. Evaluate xe x dx. x Evaluate the corresponding proper integral first, then take the limit. (You have to remember how to compute limits.) EX 3. Evaluate + x 2 dx. If the integral is improper at both ends, cut it in two and handle the two infinities separately. EX 4. For what values of p is the integral dx convergent? xp If p is big enough, then x gets small fast enough that you can add it all up and still p get something finite; if p is too small, then x gets small too slow (or even gets big, p if p < 0) and adding it all up makes. The result of this example is worth memorizing; it ll come in handy later (when we do power series). EX 5,6,7,8. Examples with discontinuous integrands. Same as before: evaluate a corresponding proper integral and take the limit. From this point on, integrands may have discontinuities at or between the limits of integration; you are expected to notice them and deal with them. Be alert. EX 9. Show that 0 e x2 dx is convergent. Sometimes the direct approach (as in previous examples) fails because you don t know the antiderivative; the comparison theorem can tell you about convergence or divergence (but not the exact value) using related functions whose antiderivatives you do know. The exact value of this particular integral can be computed with a clever trick you ll probably see in multivariable calculus next year. EX 0. Show that + e x dx is divergent. x Often a good comparison can be had by just throwing away the most complicatedlooking bit of the integrand. Math 0 Lab ET3 2009 February
0 9.3: Separable Differential Equations EX. Solve the differential equation dy dx = x2. Find the solution that satisfies the initial y2 condition y(0) = 2. Separate the variables and integrate both sides separately. Use the initial condition to find the value of C. Introduce the C at the right moment instead of just tacking it on at the end. Doing this wrong can make your solutions go deeply haywire. EX 2. Solve the differential equation dy dx = 6x 2 2y + cos y. Sometimes you can t solve for y in the final solution. EX 3. Solve the equation y = x 2 y. Notice how delicately you have to handle the absolute value here. Notice also how the +C became a multiplicative parameter. EX 4. Nothing new. EX 5. Find the orthogonal trajectories of the family of curves x = ky 2, where k is an arbitrary constant. Describe the given family with a differential equation, change the equation to describe the orthogonal curves, then solve the new equation. This part of the course makes the coolest pictures. EX 6. Mixing problem. What we understand about the physical system described is how it changes from moment to moment; such knowledge is most naturally mathematized in terms of derivatives, yielding a differential equation. Solving the equation gives you complete knowledge of the system. This is a typical situation in the sciences, which is why we care about differential equations. 2009 February Math 0 Lab ET3
.2: Series EX 5. The geometric series ar n. n= The value of an infinite series is the limit of the sequence of its partial sums. Memorize the results of example. Some geometric series are algebraically disguised; know your rules of exponents. Repeating decimals are geometric series in disguise. EX 6. The series n= n(n + ). Partial fractions isn t just for integrals. Telescoping sums are usually disguised and need some trick to expose them (like partial fractions here). Try this one: n+ n= ln n. EX 7. The harmonic series n= n. The terms get smaller and smaller as we go, but not fast enough, so their sum gets arbitrarily big. You should think this is weird. To show this series diverges, we show its partial sums get arbitrarily large. That idea is applicable to other series, but the particular method of grouping terms used in this example isn t. EX 8. Show that n= n 2 5n 2 + 4 diverges. The test for divergence: if the terms don t get small, the sum diverges. Get used to spotting the dominant pieces of functions. The idea in this example is n that, when n is big, 2 5n 2 +4 n2 = 5n 2 5. EX 9. Find the sum of the series ( n= 3 n(n + ) + 2 n Series add up like you d expect, and you can factor things out and multiply them in like you d expect, as long as everything is convergent. Math 0 Lab ET3 2009 February 25 ).
2.3: The Integral Test and Estimates of Sums EX. Test the series n= n 2 for convergence or divergence. + If you happen to recognize the term of a sum as something you d be able to integrate, use the integral test. The integral and the sum usually have different values! EX 2,3. For what values of p is the series Memorize the result of this example. Compare to 7.8, Example 4. EX 4. Determine whether the series n= n= ln n n n p convergent? converges or diverges. When applying the integral test, remember to check the conditions that the function is continuous and (eventually) positive and decreasing. EX 5,6. Approximation of sums using integrals. A quantitative version of the integral test. Chiefly useful for understanding why the integral test works at all; see the pictures on page 703..4: The Comparison Tests EX,2. Comparison examples. When the terms are positive, a series either converges to some value or diverges to. So if a series diverges, any bigger series also diverges, and if a series converges, any smaller series also converges. The chief difficulty in applying the comparison test is choosing a good series to compare to. Usually you should just throw away everything but the dominant bits (but sometimes you have to fiddle a bit). EX 3,4. Limit comparison examples. If an b n 0, then the a n are smaller than the b n (more precisely, the a n get small faster than the b n do); if an b n, then the a n are larger than the b n ; anything in between means they re roughly of the same size, so the series behave the same. 2009 February 25 Math 0 Lab ET3
3.5: Alternating Series EX. The alternating harmonic series ( ) n+. n n= Recall that if a series converges, then its terms go to zero, but the converse is false: just because the terms go to zero doesn t mean the series converges. (See.2, example 7.) The alternating series test gives a limited partial converse. EX 2. The series n= ( ) n 3n 4n. If the alternating series test fails because the terms don t go to zero, the Test for Divergence applies. (If it fails because the terms don t get smaller, all kinds of things can happen. That s a good exercise: find an example of an alternating series whose terms go to zero but which diverges; find another example whose terms don t get smaller but which nevertheless converges.) EX 3. The series ( ) n+ n 2 n 3 +. n= Doing the first part of the alternating series test (the part about the terms getting smaller) might involve just a little algebra, but it might be easier to use differential calculus. EX 4. Find the sum of the series ( ) n n=0 n! correct to three decimal places. Alternating series are easy to estimate, since the partial sums keep bracketing the sum of the series. (See the picture on page 70.).6: Absolute Convergence and the Ratio and Root Tests ( ) n EX. The series n= n 2 is absolutely convergent. This example just shows what the definition says. Math 0 Lab ET3 2009 March 4
4 EX 2. The alternating harmonic series ( ) n is only conditionally convergent. n n= The following theorem says that if it s absolutely convergent then it s convergent; this example shows the converse is false: just because it s convergent doesn t mean it s absolutely convergent. (So this example shows that the notion of absolute convergence is really a new thing.) EX 3. Determine whether n= cos n n 2 is convergent or divergent. Our first example of a series with both positive and negative terms but which isn t alternating. EX 4. Test the series ( ) n n3 for absolute convergence. 3n n= The ratio test: if a n+ a n L, then (ignoring the absolute values for a minute) a n+ La n, so the series resembles a geometric series with common ratio L. This is why such series converge if L < and diverge if L >. (If L =, the series is close to the boundary between convergent and divergent series, and it has enough wiggle room to go either way.) In this series, taking the ratio of successive terms works well because most of the 3 n part cancels. In general, exponential things play nicely with the ratio test. EX 5. Test the convergence of the series n= n n n!. In the ratio test, factorials in successive terms mostly cancel (which is good), but something like n n doesn t (so we need some other trick to figure out the limit of that part of the ratio). This example is a little silly; as Stewart notes afterward, we can see pretty easily that the terms of this series don t go to zero, so the series diverges. The only reason to give this example is to remind you that ( + n )n e. EX 6. Test the convergence of the series n= ( ) 2n + 3 n. 3n + 2 The ratio test would be a bad choice here because stuff in successive terms won t cancel. (If you don t see why, try it.) The root test plays better with nth powers when the base also involves n. 2009 March 4 Math 0 Lab ET3
5.8: Power Series EX 5. What values of x various series converge for. The methods are the usual ones for determining whether a series converges chiefly the ratio and root tests but with an unknown x in the middle of things. So the series convergence will (often) depend on x. There are three possible outcomes: the series converges only at its central value; it converges when within some distance of its central value and diverges when further away; or it converges for all x. In the middle case, you have to check the endpoints separately..9: Representations of Functions as Power Series EX 3. Express some functions as the sum of a power series and find their intervals of convergence. Some good examples of a general problem-solving technique: make what you have look like something you know. For power series, the usual target is x. Some fluency with algebraic manipulations is required to pull this off. The force-factoring trick in example 2 should be familiar; see 7.3, example. Multiplying by a power of x is easy in power series. EX 4. The derivative of the Bessel function. You can differentiate power series term-by-term. This works inside the interval of convergence, and leaves the radius of convergence the same, but the series you get for the derivative might diverge at the endpoints even if the original didn t. (Exercise: find an example of that.) EX 5 7. Express, ln( x), and arctan x as power series. ( x) 2 How to use term-by-term differentiation and integration to get power series for some more functions. 0.5 EX 8. Evaluate dx as a power series and approximate + x7 + x 7 dx within 0 7. A straightforward combination of techniques from previous examples: get a power series for the integrand (like in example ), integrate term-by-term (like in examples 6 and 7), and approximate the series (like in.5, example 4). Math 0 Lab ET3 2009 March 0
6.0: Taylor and Maclaurin Series EX. Find the Maclaurin series of the function f(x) = e x and its radius of convergence. Shows direct use of the formulas derived in the text. Functions with simple derivatives have simple power series. EX 2. Prove that e x is equal to the sum of its Maclaurin series. The usual use of Taylor s inequality. There are some weird functions which have Taylor series but aren t equal to them, but most functions encountered in real life are better behaved. EX 3. Find the Taylor series for f(x) = e x at a = 2. You can find a Taylor series about any point. One use of Taylor series is that truncating them gives polynomials that approximate your function; but such an approximation is only good if the omitted terms are small, that is, if (x a) n is small for large n, that is, if x a. So you have to expand around a point in the middle of the values you re interested in. EX 4,5,7. Just like examples 3. EX 6. Find the Maclaurin series for f(x) = x cos x. Taylor series are power series, so the lessons from the previous section apply. This example repeats the lesson of example 3 there. EX 8,9. The binomial series. EX 0. Just like example 8 of the previous section. EX. Evaluate lim x 0 e x x x 2. A cute use of Taylor series for evaluating limits. Read the remark on example 3 again, then consider: what should you do for lim x a? EX 2. Find the first three nonzero terms in the Maclaurin series for e x sin x and tan x. Multiplying and dividing power series is much like multiplying and dividing polynomials. 2009 March Math 0 Lab ET3
7 0.: Curves Defined by Parametric Equations EX. Sketch and identify the curve defined by the parametric equations x = t 2 2t and y = t +. The only advantage of plotting points is that it s mechanical and so requires no thought. The disadvantages are that it takes a lot of time, it s error-prone, and it doesn t reliably show all the curve s behaviour. Eliminating the parameter, when possible, brings us back to the familiar world of equations in x and y. EX 2 4. Parametric equations for circles. Be ready to recognize this cos, sin form. A parametric equation doesn t just describe a curve, but how it is traversed. Translating a curve is easy in parametric coordinates. EX 5. Sketch the curve with parametric equations x = sin t, y = sin 2 t. A good combination of eliminating the parameter and straight-up reasoning about the equations. This is oodles better than plotting points. EX 7. The cycloid. A good example of mathematizing something: the description of the situation gives rise to a picture (figure 4) which gives rise to equations describing the situation. You should be able to do this yourself (though it might well take some time). 0.2: Calculus with Parametric Curves EX,2. Tangents to curves given by parametric equations. dy dx = dy/dt. Otherwise such problems use the same ideas as they did in Math 00. dx/dt Math 0 Lab ET3 2009 March 8
8 (I think you ll be skipping the next two topics now and coming back to them later.) EX 3. Areas under curves given by parametric equations. y dx = y(t)x (t)dt. Draw a picture before trying this, to make sure that the idea of area under the curve really makes sense for the curve at hand. (If the curve fails the vertical line test, for example, it doesn t make sense.) EX 4 6. Arclength for curves given by parametric equations, and surface area for surfaces formed by revolving such curves. ds = (x (t)) 2 + (y (t)) 2 dt (which is much more symmetrical than the corresponding formula when y is a function of x, namely ds = + (y (x)) 2 dx). 0.3: Polar Coordinates EX 3. Using polar coordinates to describe points, and converting between Cartesian and polar coordinates. r is distance from the origin, θ is direction. When converting from Cartesian coordinates to polar, use the signs of x and y to determine the quadrant, which gives the range of possible θ. (This can t be told just from the fact tan θ = y x.) EX 4 6. Using polar coordinates to describe curves. This takes some getting used to. About as much getting used to as Cartesian coordinates did, the first time you met them. (Remember that?) EX 7,8. More curves in polar coordinates. This trick of sketching r and θ on Cartesian axes first is pretty good; it will let you get a (somewhat vague) sense of how the curve works in a hurry. EX 9. Tangents in polar coordinates. Polar equations can (usually) be turned into parametric ones via x = r cos θ and y = r sin θ. The resulting formula for dy dx can be memorized, but it s probably easier just to remember how to derive it. 2009 March 8 Math 0 Lab ET3
9 0.4: Areas and Lengths in Polar Coordinates EX. Find the area enclosed by one loop of the four-leaved rose r = cos 2θ. A = b a 2 r2 dθ. Figuring out the right values of a and b can be tricky. In this example, you could use the fact that the loops start and stop at the origin, and find θ such that r = 0. EX 2. Find the area of the region that lies inside the circle r = 3 sin θ and outside the cardioid r = + sin θ. Recall that, when finding the area above one curve and below another (see 6.), you have to figure out where the curves cross and which is above which where. Same thing here: you have to find out when one curve lies outside the other. Algebra required. Since polar equations usually involve trigonometric functions, familiarity with trigonometric equations and inequalities also required. EX 3. Find all points of intersection of the curves r = cos 2θ and r = 2. A sneaky thing about polar coordinates. EX 4. Find the length of the cardioid r = + sin θ. (Not this week, though.) b ( ) dr 2 L = r 2 + dθ. Derive this formula a few times yourself; that s the best a dθ way to memorize it. 2.: Three-Dimensional Coordinate Systems 2.5: Equations of Lines and Planes (Covered in Math 02.) Math 0 Lab ET3 2009 March 25