1 Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.-3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number of prctice questions for you to work on. The third section give the nswers of the questions in section. Review 7.4: Integrtion of Rtionl Functions by Prtil Frctions Rtionl functions consist of frctions of polynomils. We cn split rtionl functions into simpler pieces by prtil frctions. Remember tht prtil frction decompositions re bsed on liner nd qudrtic fctors in the denomintor. For ech liner fctor, we hve term with constnt in the numertor nd the fctor in the denomintor. For ech irreducible qudrtic, we hve term with liner function in the numertor nd the qudrtic in the denomintor. For exmple, x x + (x )(x + )(x + x + 3) = A x + B (x + ) + Cx + D x + x + 3. () We just need to determine the vlues of A, B, C nd D. This is done by plugging in vlues for x: You need to plug in s mny numbers s you hve constnts. Using some numbers, (like - nd in this cse) mkes your life esier, but ny four numbers will do. Notice tht if we multiply eqution by the denomintor on the left side, we get x x + = A(x + )(x + x + 3) + B(x )(x + x + 3) + (Cx + D)(x )(x + ). () Letting x = in eqution gives 8 = 5B, so B = 8/5. Letting x = gives = 5A, so A = /5. Letting x = 0 gives = 5A 3B D. Knowing A nd B helps us to find D. Finlly, if x =, 4 = 9A 6B + C D, llowing us to solve for C. Remember tht repeted fctors must give repeted terms with incresing exponent in the denomintor. For exmple, x 3 + x + x 5 (x ) 3 (x + 9) = A x + B (x ) + C (x ) 3 + Dx + E x F x + G (x + 4) Finlly, remember prtil frctions only works if the degree in the numertor is less thn the degree in the denomintor. Otherwise, you need to divide nd use prtil frctions on the reminder. 7.5 Strtegy for Integrtion You my hve noticed tht we hve relly only used two techniques of integrtion in chpter 7: substitution nd integrtion by prts. Everything else is just mnipultion of those two techniques. Luckily, most integrls tht require specific technique hve ptterns tht help remind us. Those ptterns hve been discussed erlier. You re expected to know them. A generl strtegy for ttcking unknown integrls is s follows:. Cn substitution be used to simplify the integrl? If so, do this first. (Note tht in this section, substitution is often just the first step. After the substitution, some other technique needs to be pplied.). If you cnnot use substitution, or you hve used it but the integrl is still not simple enough, look for ptterns. () If the integrl is product of trig functions, use the ptterns we lerned in 7. to tckle it. (b) If the integrl hs sum or difference of squres in the integrnd, use trig substitution. (c) If the integrnd is rtionl function, use prtil frctions. 3. If you cnnot see one of the bove ptterns, nd cnnot use substitution, try integrtion by prts.
2 7.7 Approximte Integrtion In this section we concentrte on two things: How cn we efficiently pproximte the solution to definite integrl, nd how cn we pproximte the error. You re expected to know how to clculte the following pproximtions of definite integrl f(x) : () the left nd right hnd sum Left Hnd Sum: L f = x n k= f(x k ) Right Hnd Sum: R f = x n k= f(x k) (b) the Trpezoid rule T f = x ( ) n f() + f(b) + f(x k ) k= (c) the Midpoint rule n ( ) xk + x k M f = x f k= (d) Simpson s rule S f = x 3 [f(x 0) + 4f(x ) + f(x ) + 4f(x 3 ) + f(x 4 ) + + 4f(x n ) + f(x n )]. Remember, n must be even to use Simpson s rule. It is unlikely tht you will need to clculte the left or right hnd sums, but you will still need to know bout them. You re expected to know how ech of these rules behve bsed on stndrd properties of the function (monotonicity, concvity, etc). For exmple, it is well known tht the left hnd sum is lower bound nd the right hnd sum is n upper bound of the definite integrl of incresing functions. Wht behvior is true for Midpoint nd trpezoid? You re expected to know the error estimtes of Trpezoid, Midpoint nd Simpsons: Trpezoid with f (x) K on [, b]. E T K(b )3 n Midpoint with f (x) K on [, b]. E M K(b )3 4n Simpson s with f (4) (x) K on [, b]. E S K(b )5 80n 4 You cn be expected to use the error estimtes in one of two wys: to estimte the error of the clcultion for prticulr vlue of n, or to find vlue for n tht gives n error no more thn some stted vlue. 7.8 Improper Integrls Remember, there re two types of improper integrl: Infinite length: Integrls of the type Unbounded integrnd. Integrls of the type where f hs n infinite discontinuity t. f(x), f(x), c f(x). f(x)
3 Remember, in ech type, there is problem tht definite integrl cnnot hndle. We remove the problem by turning it into limit: c f(x) = lim b f(x) = lim b Some things to remember when clculting improper integrls: c f(x) f(x). Do not forget to set up n improper integrl s limit. You will likely hve points deducted if you do not. It is the only wy for the grder to tell tht you know wht you re doing. Wtch out for infinite discontinuities in the middle of the intervl. You must split the integrl t the discontinuity in tht cse. 8. Arc Length To find the length of curve given by f(x), x b, the formul is s = + (f (x)). Notice tht there is reson tht this section is not in chpter 6. Arc length integrls cn be difficult to solve. You my need ny of the techniques from Chpter 7 to clculte the rc length. If the integrl is complicted, you my wish to simplify it first. for exmple, when f is frction, you my wish to rewrite + (f (x)) s single frction before you proceed. 8. Surfce Are Here re the importnt steps to keep in mind when solving this problem: First sketch the curve nd identify the xis of rottion. Choose vrible, either x or y, to serve s our independent vrible. If the curve is given s function of x (e.g., y = x + ), then we will wnt to choose x s our independent vrible, while if the curve is given s function of y (e.g., x = y 3 y + ), we will wnt to choose y s our independent vrible. This is the vrible with respect to which we will be integrting. If the curve is given implicitly (e.g., x + y = ), then we my choose either vrible nd solve for the other vrible. Next consider the circumferences of the circles formed by revolving points on the curve bout the xis of rottion. The circumference of such circle will be πr, where r is the distnce to the xis of rottion. This circumference should be expressed s function of either x or y, whichever one we chose in the previous step to be our independent vrible. The generl formul for the re of surfce of revolution of this type is Exmples: S = (circumference) ds where circumference is wht we found bove, nd ds is either whether x or y is our independent vrible, respectively. Rotting f(x), x b bout the x xis: Rotting f(x), x b bout the y xis: b Rotting f(x), x b bout the xis y = c: Rotting f(x), x b bout the xis x = d: + πf(x) + (f (x)). πx + (f (x)). b Exercise: Write the pproprite formuls for function of y. π f(x) c + (f (x)). π x d + (f (x)). ( ) dy or + Note tht you my need the sme techniques to solve these integrls s you do in section 8.. ( dy ) dy, ccording to
4 8.3 Applictions to Physics nd Engineering Fluid pressure The clcultion of fluid pressure on plte verticlly suspended in liquid is ωd(h)l(h) dh where D(h) is the depth t h, nd L(h) is the width of the plte t h. Centroid There re three types of problems tht you will need to be ble to clculte. Using the property of sums of moments to be ble to find centroids nd center of mss. (Used in point msses, nd in regions where the re is esily clculted.) Finding the moments with respect to the x nd y xes, nd the centroid of region bounded by two functions. For exmple, if the region is bounded by then we hve: Since the re of the region is the centroid is given by f(x) y g(x), x b, Moment bout the y xis:m y = Moment bout the x xis:m x = A = x = M y A, (f(x) g(x)), y = M x A. x(f(x) g(x)) (f (x) g (x)). finding the center of mss of plte. Here, we hve density. (mss is density times volume). If the density is constnt, the moments re just the clcultions bove times the density. The mss of the object is lso the re times the density. The center of mss, however, is the sme s the clcultion for the centroid. In the cse of density tht is not constnt, the sitution is more complicted. However, since the homework only hs constnt density, tht is ll tht need concern us for this exm. You my need to rework the bove equtions for regions contined by functions of y. Geometric properties of the centroid We now turn to some properties of the centroid which cn simplify certin geometricl problems.. Centroids of tringles re given geometriclly If you re trying to find the centroid of tringle, you cn do the following: Mesure /3 long the line connecting vertex to the midpoint of the opposite side. Tht s the centroid! (If you drw the lines from ech vertex to the midpoint of the opposite side, they ll intersect in the centroid.). Use symmetry The centroid will lwys lie on line of symmetry (if the object possesses one). For exmple, suppose we hve tringle with vertices t (0, 0), (, 0) nd (, 5). This is n isosceles tringle. By the symmetry, it is cler tht x =. From the lst item, since the line from the vertex bove to the midpoint is verticl, we see tht y = 5/3. 3. Moments Add! If you re trying to find the moment of complicted region, you cn split the region into simpler regions nd dd the moments together. For exmple, suppose circle of rdius nd squre of length re plced side by side. Where is the centroid of the system?
5 The centroid of the circle is (, ). Thus, the moment of the circle bout the y xis is - times the re of the circle: π. Similrly, the moment of the circle bout the x xis is π. Since the centroid of the squre is t (/, /), the moments of the squre bout the y nd x xes re both / (since the re of the squre is ). Thus, the totl moment bout the y xis is π + /. Also the totl moment bout the x xis is π + /. Hence, the centroid is given by π + / x = π + nd y = π + / π Other clcultions look like moment clcultions Recll tht when we developed the clcultions for force due to fluid pressure, we took smll strip nd multiplied the re by ω depth. Tht is like clculting moment! It should be no surprise, therefore, tht if the force due to fluid pressure on the (submerged, verticl) plte is (weight density of liquid) (depth of centroid) (re of plte). We lso hve the first theorem of Pppus for the clcultion of volumes of rottion: Volume = π (distnce from centroid to rottion xis) (re). You should only use these formuls either when you lredy hve the centroid, or when it is esy to find. For exmple, suppose we wish to find the force on the side of circulr plte of rdius feet tht is submerged in wter to point 5 feet from the top of the plte. Since the centroid of the plte is 7 feet from the surfce, the force is given by F = π = 437.5πlb. Notice how much esier the clcultion becomes! Tht is becuse the centroid ws esy to find. 8.5 Probbility This section is concerned with the probbility of continuous rndom vrible. A continous rndom vrible X is described by probbility density function f(x) with the following properties: ) f(x) 0 on (, ). b) f(x) =. With probbility density function, you cn perform the following clcultions: ) Find the probbility tht X is between A nd B: P (A < X < B) = B f(x) A b) Clculte the men: µ = xf(x) c) Clculte the medin: Find m so tht m f(x) = d) Clculte the stndrd devition: σ = (x µ) f(x) Questions Try to study the review notes nd memorize ny relevnt equtions before trying to work these equtions. If you cnnot solve problem without the book or notes, you will not be ble to solve tht problem on the exm.
6 . 6 x 3 +x +x. x +x 5 x 3. x x 3 +x 4. x 3 x 4 +x x x 4 cos x 6. π 0 tn( x )] [Hint: Use rtionlizing substitution] [Hint: Use the substitution t = 7. Estimte 4 0 e x with midpoint, trpezoid, nd simpson for n=4 nd Clculte bound on the error for the trpezoid clcultion in the lst question (n=4 only). 9. How lrge does n need to be in order for midpoint to hve n error no lrger thn 0 5? 0. How lrge does n need to be in order for Simpson s to hve n error no lrger thn 0 5?.. 3. For questions to 7, evlute the integrl, or show tht it diverges. x ln x ln x x 4 4. ln x 0 (x + ) 3 5. π sec x π/ 6. ln x x 4 For questions 8 to 9, use the Comprison Theorem to show whether the improper integrl converges or diverges. tn x x + e x 3 x 0. Find the rc length function for the curve y = x with initil point (0, 0).. Find the rc length function for the curve y = x 3 with initil point (0, 0).. Find the rc length of the curve C : y = 3 + cosh(x) for 0 x. 3. Find the rc length for the curve y = x x + sin ( x) for 0 x. In questions (4) to (8), find the re of the surfce formed by revolving the curve bout the specified xis: 4. y = x +, x 3, bout the y-xis. 5. x = sin y, 0 y π, bout the y-xis. 6. y = + x, x, bout the line y =. 7. y = x 3 +, 0 x, bout the line y =. x y 9 =, bout the x-xis. 9. A circulr pipe of 4m dimeter hs plte tht regultes the flow. When the plte is in position, the flow is completely cut off. Wht is the force on the plte in this position, if the pipe is full of wter on one side? 30. Find the force due to fluid pressure on one side of plte in the shpe of the upper hlf of regulr hexgon of dimeter 8 feet if the hexgon is sitting verticlly in wter with the (flt) top of the hexgon t depth of 5 feet. 3. A cylindricl brrel with height m nd dimeter m is filled to the brim with oil. If the weight density of oil is 900 kilogrms per cubic meters, wht is the force due to fluid pressure on the side of the brrel? 3. The msses m i re locted t the points P i. Find the totl moments M y nd M x of the system, nd find the center of mss of the system. m = 4, m =, m 3 = 3 P = (, ), P = (, 3), P 3 = (4, ) 33. Let R be the region bounded by f(x) = sin(x) nd g(x) = cos(x) for x [π/4, 5π/4]. Sketch the region R, nd find the centroid (center of mss) of R. 34. Let R be the region bounded by f(x) = e x nd g(x) = x for x [0, ]. Sketch the region R, nd find the centroid (center of mss) of R. 35. Let R be the region bounded by the curves f(x) = x nd g(x) = x. Sketch the region R, nd find the centroid (center of mss) of R. 36. Let R be the region consisting of two djcent circulr discs described by the equtions (x ) +y = nd (x 3) + y =. Use the theorem of Pppus to find the volume of the solid obtined by rotting R bout the y-xis. 37. Let R be the cross shpe formed by tking squre of side length 4, nd cutting out of ech corner smller squre of side length. Use the theorem of Pppus to find the volume of the solid obtined by rotting R bout one of its originl sides.
7 38. Find the force due to fluid pressure on verticl plte in the shpe of n equilterl tringle ( feet on side), with the lower side prllel to the surfce, nd the upper vertex t depth of 5 feet. 39. Find the centroid of the following object: (ech tic mrk is one unit) For questions 40 to 4, ) determine the vlue for c tht mkes f(x) probbility density function, b) find P (A < X < B) nd P (X > B), c) find µ, nd d) find σ. 40. f(x) = cxe x on [0, ), A = 0, B =. 4. f(x) = c (+x ) 3 on (, ), A =, B =. 4. f(x) = cx( x) on [0, ], A = 0, B = /.
8 Answers. 6 ln x 6 ln x x+ + C. x 3 ln x + 5 ln x + + C 3. ln x + ln x + + tn x + C 4. (ln x + + x + ) + C 5. (ln 5 ln 3 + ) π 7. n = 4: Midpoint Trpezoid Simpson n = 8: Midpoint Trpezoid Simpson f (x) < on [0, 4]. E < f (4) (x) <. n = 5.. Diverges Diverges Diverges. 8. Converges. Mke the comprison tn x x < π/ x. 9. Diverges. Mke the comprison + e x 3 x > 3 x. 0. x + x + ln ( x + + x). L(x) = ( + 9x) sinh 4. 8π 5 5. π( + ln( + )) π π ln ( + 5 ) π 7. 7 (0 0 ) ( ( 8. π ln 9. 8πω = 46300N 3+ )) ω( ) = lbs. 3. πω = π = N 3. M y =, M x =, ( x, ȳ) = (4/3, /9) ( ) 3π 33. 4, 0 ( ) e 6, 3(5e 7) 0(3e 4) 35. (0, ) 36. 8π π ( ) 3 = lb 39. ( + 64π π + 64π, ) = ( 0.05,.58) π 40. c = 4, 3e, 3e,, / 4. c = 8/(3π), , , 0, / 6 4. c = 6, /, /, /, / 5.