MATH 230 CALCULUS II OVERVIEW This overview is designed to give you a brief look into some of the major topics covered in Calculus II. This short introduction is just a glimpse, and by no means the whole picture. Think of it as a preview to a movie. Where we left off from Math 229, Calc I Recall that the integral f(x)dx is the limit of the sum of rectangular approximations b a n f(x i ) x i= as the number, n, of the rectangles goes to infinity. Here the interval of integration [a, b] is broken into n equally spaced subintervals of width x = b a n, with endpoints a = x 0, x, x 2,..., x n, x n = b. The ith starred point x i lies in the ith subinterval [x i, x i ]. This stuff looks pretty complicated, but two important facts simplify our life considerably: Fact. If the function f(x) is continuous on the interval [a, b], then the limit of the rectangular (or Riemann) sum and hence the integral exists. Fact 2. If the function f(x) has an antiderivative F (x), that is, if F (x) = f(x), then the integral is just b a f(x)dx = F (b) F (a). Typeset by AMS-TEX
2 MATH 230 CALCULUS II OVERVIEW Fact says that you can always integrate a continuous function while Fact 2 asserts that integrals are antiderivatives. Using Integrals to Calculate Volumes At the end of Calculus I you saw how to use an integral to evaluate the area between two curves; you just integrate the top curve minus the bottom curve. The hard part often is deciding which curve is on top. We extend this idea to evaluate the volume of certain solid figures. For example suppose we want to know the volume of a loaf of bread. Run the loaf through a bread slicer, obtaining n slices of bread (suitable for toasting). The volume of each slice is approximately A(x) x where A(x) is the cross sectional area of the bread at each x-coordinate and x is the width of each slice. The volume of the bread is approximately n A(x i ) x. i= As the number of slices, n, gets larger and larger and consequently, the slices themselves get wafer thin the above sum gets closer to the true volume of the loaf of bread. We define the exact volume to be the limit of this sum as n. The point is that the limit of this (Riemann) sum is by definition the integral of A(x), hence the volume of our loaf of bread becomes V = b a A(x) dx. The cross-sections are typically triangles, rectangles, disks, or washers (the region between two concentric circles). This slicing method works for finding the volumes of many familiar objects in geometry: cylinders, pyramids, cones, spheres, doughnuts. To obtain a doughnut mathematically, rotate a small circle around a big circle kind of like the round tunnel surrounding Fermilab.
MATH 230 CALCULUS II OVERVIEW 3 The Natural Logarithm and the Exponential Function Since integrals are antiderivatives, we get an easy formula for integrating powers. x n dx = xn+ n + + C. Unfortunately the formula fails for n = because you end up dividing by zero, a mathematical sin. So how do you integrate the reciprocal function f(x) = x? Since the /x curve is continuous away from the origin, Fact above guarantees that for each positive number x, there is a number y whose value is precisely the area under the reciprocal curve from to x. We give the number y a name and call it the natural logarithm of x. Note that the natural log of x, or in symbols, ln x, is defined by an integral ln x = x t dt. By the fundamental theorem of calculus the derivative of this function is just the inside reciprocal function evaluated at x, that is, d x dx t dt = x. We have thus found a function whose derivative is /x. But did we actually find it or did we just name it? Mathematics is full of examples where we solve difficult problems by simply giving the solution a name. For example, any third grader knows the number whose square is 25 is 5. The number whose square is 26 is harder, and to solve the problem we give it a name, the square root of 26 and a symbol 26. Suppose an algebra problem asks you to find Mary s age. You can always say let M be Mary s age. Then if your teacher asks you What is Mary s age? you can answer: M. Nice try, but your teacher, no doubt, wants you to find the actual number represented by M. Similarly, if you need to use the square root of 26, say, to compute the height of a building, then you need a calculator to evaluate the first 0 decimal digits of 26. There are numerical methods for finding the first
4 MATH 230 CALCULUS II OVERVIEW 0 digits of the area under the reciprocal curve from a = to b = 2, that is, the first ten digits of ln(2). Why are areas under the reciprocal curve called logarithms? Didn t logarithms have something to do with exponents? The answer is that the ln function is called a logarithm because it acts like a logarithm. By this I mean that one of the fundamental facts about the log function base 0 is log(xy) = log(x) + log(y). It turns out that the function ln obeys the same rule ln(xy) = ln(x) + ln(y). Just as the log base 0 function is related to the power function y = 0 x, so is the natural log function is related to the exponential function y = e x, where the number e is approximately 2.78. Both functions ln and e x are standard keys on a scientific calculator. They are used to predict population growth and radioactive decay. When Integrals Get Tough The good news about integrals is that, like derivatives, there is a sum rule [ (f + g) = f + g], a constant rule [ (cf) = c f], and the power rule. The bad news is that there is no product rule, quotient rule, or chain rule for integration. Substitution might remind you of the chain rule, but you need to be lucky. For example, suppose you wish to integrate x x 2 dx. You can try substituting u = x 2, so that du = 2x dx or dx = du 2x.
MATH 230 CALCULUS II OVERVIEW 5 Making this substitution gives x dx = x 2 = x du u 2x 2 u /2 du substitute for x and dx cancel the two x s = u /2 + C use the power rule. That was easy, but we were lucky that the x s canceled. Suppose you were asked to integrate dx. x 2 Now when you try the substitution u = x 2, you get stuck. It turns out that the only way to solve this integrals involves trig, not an obvious fact at all. The trick is to make the substitution x = sin u. Then Our integral can now be solved: dx = x 2 = dx = cos u du. cos u du sin 2 u substitute for x and dx cos u cos2 u du since cos2 u + sin 2 u = cos u = cos u du = du cancel the square root and square cancel cos u = u + C. Can we write u as a function of x? Since x = sin u, it follows that u = sin x; so the above integral involves the inverse sin of x, that is, u is the angle whose sine is x. Mathematicians have found several ingenious methods for solving hard-to-do integrals. A large part of the middle of this course involves discussing these techniques.
6 MATH 230 CALCULUS II OVERVIEW Sequences and series What is the value of 2 + 4 + 8 + + 2 n +? This problem first got air time over two thousand years ago thanks to a mathematical prankster named Zeno, who went around confusing people with his riddles. Zeno s Paradox, roughly translated, is In order to travel from your couch to the door, you must first get to the halfway point, from there you must travel to the second halfway point, then to the third halfway point, and so on. Since there are infinitely many halfway points, you can never reach the door. Zeno, being male, expounded the ultimate guy belief: if you can t get off the couch, then you might as well lay back and watch some football. (Of course, somebody needs to bring the snacks.) It is clear that the infinite sum will never exceed one: when you get to the first halfway point, you have one half of the distance left to go; when you get to the second halfway point, you have one fourth of the original distance to go, etc. As an equation, this observation becomes 2 + 4 + 8 + + 2 n = 2 n. Now take limits as n tends towards infinity. (Isn t it funny how many concepts in calculus involve limits?) Since the sum of all the terms lim n 2 n =, 2 + 4 + 8 + + 2 n + =. This is an example of an infinite series, that is a sum with infinitely many terms. It may surprise you to learn that you can add together infinitely many terms and
MATH 230 CALCULUS II OVERVIEW 7 still get a finite sum. We do this all the time with decimal numbers. The number π = 3.459 can be written as an infinite sum π = 3 + 0 + 4 00 + 000 + 5 0000 +. When a series has a finite sum like this, we say the series converges. It turns out that some series do not converge, even though their terms gets smaller and smaller. For example, the sum + 2 + 3 + 4 + 5 + + n + turns out to be. In such cases we say the infinite series (or sum) diverges. The game is: given an infinite series, does it converge or diverge? You will learn general techniques for making this decision for rather complicated series. One of the major applications of series, by the way, is to evaluate numerical functions. How does your calculator compute the cosine of an angle (in radians)? There is no 0 place trig table stored in memory. The answer is that it adds enough terms of an infinite series that converges to the cosine function to obtain the desired accuracy.