MATH 7.57 Exam Solutios February, 6. Evaluate (A) l(6) (B) l(7) (C) l(8) (D) l(9) (E) l() 6x x 3 + dx. Solutio: D We perform a substitutio. Let u = x 3 +, so du = 3x dx. Therefore, 6x u() x 3 + dx = [ ] 9 u() u du = l u = l 9 l = l 9.. Cosider the regio i the first quadrat bouded by x = y, y = 3, ad the y-axis. Which itegral below gives the volume of the solid obtaied by rotatig this regio about the x-axis? (A) π (B) π (C) π (D) π (E) π 9 9 9 3 3 (9 x) dx x dx x(3 x) dx y(3 y ) dy y dy Solutio: A We first fid the itersectio poit betwee x = y ad y = 3, which is easily see to be (x, y) = (9, 3). The regio i questio the sits above the iterval x 9 ad betwee the curves y = x ad y = 3. We the eed to use washers to fid the volume of the solid obtaied by rotatig the regio aroud the x-axis. We see that for a give x with x 9, the outer radius is 3 ad the ier radius is x. Thus, V = π 9 (3 ( x) ) dx = π 9 (9 x) dx.
π 3. Evaluate x si(x) dx. (A) π + (B) π (C) π (D) π (E) π Solutio: C We use itegratio by parts, pickig u = x ad dv = si(x) dx. The du = dx ad v = cos(x). Thus, x si(x) dx = x cos(x) + cos(x) dx = x cos(x) + si(x), ad so π x si(x) dx = [ ] π x cos(x) + si(x) = ( pi( ) + ) ( + ) = π.. A meter cable has a total mass of /(9.8) kg ad a costat mass desity. It is iitially hagig from the top of a tall buildig. How much work (i Joules) is doe agaist gravity whe liftig the cable to the roof of the buildig? (Assume that the acceleratio of gravity is 9.8 m/s.) (A) (B) (C) 3 (D) (E) 5 Solutio: B We see right away that the mass desity δ of the cable is δ = (9.8)() kg/m. A piece of the cable of ifiitesimal width dx that is a distace x from the top of the buildig, the has a mass of m = dx kg, (9.8)() ad so the force of gravity actig o this piece is F = mg = ()(9.8) (9.8)() dx = dx N. 5 Our piece at x will travel a distace x to get to the top of the buildig, so [ ] x W = 5 x dx = = 5 5 = J.
5. Let F (x) = (A) 8 8 (B) 8 (C) (D) 8 x (E) t3 + t dt. What is F ()? Solutio: C From the Fudametal Theorem of Calculus we kow that F (x) = x3 + x. Therefore, F () = 8 + =. 6. Cosider the followig limit of Riema sums: lim [ + + + + + + + Which defiite itegral below is equal to this limit? (A) (B) (C) (D) 3 3 x dx + x dx ( + x) dx x dx (E) Noe of the above ] ( ). Solutio: D Whe we look at the expressios uder the square-roots, we see that each is more tha the precedig oe, so the legth of the iterval of itegratio is. If we let f(x) = x, the the limit is [ ( lim f() + f + ) ( + f + ) ( )] ( ) + + f +, ad so we are itegratig f(x) over the iterval that starts at ad eds at + = 3. 3 Thus the limit is x dx. 3
7. Let R be the regio i the first quadrat that is below the graph of y = x x+6 ad above the graph of y = x. Fid the area of R. Solutio: We fid the itersectio poits betwee the two parabolas: x = x x + 6 yields the equatio x + x 6 = x + x 8 = (x + )(x ) =. Therefore the parabolas itersect whe x = ad x =. Sice the regio R is i the first quadrat, it sits above the iterval x. The top curve is y = x x+6, ad the bottom curve is y = x, so the area of R is A = ( x x + 6 x ) dx = = ( x x + 6) dx [ 3 x3 x + 6x = 6 3 8 + 3 = 56 3. 8. Fid the average value of the fuctio f(x) = si(3x) cos(x) over the iterval [, π ]. Solutio: The average value of f o [, π ] is ] We use the idetity π π/ si(3x) cos(x) dx. si(a) cos(b) = (si(a B) + si(a + B)). so π π/ π/ si(3x) cos(x) dx = (si(x) + si(x)) dx π = [ π cos(x) ] π/ cos(x) = [ π cos(π) cos(π) + cos() + ] cos() = [ π + + ] = π.
9. Evaluate the followig itegrals. (a) x 3 x + dx (b) cos (x) dx (c) e x si(x) dx Solutio: (a) We use substitutio. Let u = x +, so du = x dx. Also x = u. Therefore, x 3 x + dx = x x x + dx = (u ) u du = (u 3/ u / ) du = 5 u5/ 8 3 u3/ + C = 5 (x + ) 5/ 8 3 (x + ) 3/ + C. (b) We use the idetity that cos (θ) = ( + cos(θ)), ad so cos (x) dx = ( + cos(x)) dx = x + 8 si(x) + C. (c) We use itegratio by parts with u = e x ad dv = si(x) dx, so du = e x dx ad v = cos(x). Therefore, e x si(x) dx = e x cos(x) + e x cos(x) dx. We eed to perform itegratio by parts agai, with u = e x ad dv = cos(x) dx (so du = e x dx ad v = si(x)), ad we obtai e x si(x) dx = e x cos(x) + e x si(x) e x si(x) dx. Movig the itegral o the right to the left side, we have e x si(x) dx = e x cos(x) + e x si(x) + C, ad so e x si(x) dx = ex cos(x) + ex si(x) + C. 5
. Let R be the first regio i the first quadrat to the right of the y-axis that is bouded above by the graph of y = cos(x) ad below by the graph of y =. I each part below, set up a itegral for volume of the give solid (but do ot evaluate). It may help to first sketch the regio R. (a) The solid obtaied by rotatig R about the x-axis: (b) The solid obtaied by rotatig R about the lie x = : (c) The solid whose base is R ad whose cross-sectios perpedicular to the x-axis are squares: Solutio: The graph of y = / is a horizotal lie, ad it itersects y = cos(x) whe cos(x) = /, so whe x = π. Thus R is the regio that is above the iterval x π ad is bouded o the bottom by y = / ad o the top by y = cos(x). (a) Rotatig about the x-axis we eed to use washers. The outer radius is cos(x) ad the ier radius is /, so V = π π/ ( cos (x) ) dx. (b) Rotatig about the vertical lie x =, we eed to use cylidrical shells. For a shell startig at some x with x π, the radius is x + ad the height is cos(x) / x. Therefore, V = π π/ ( (x + ) cos(x) ) dx. (c) We eed to itegrate the areas of the cross-sectios, which at a give x is a square of side-legth cos(x) /. Thus, V = π/ ( cos(x) ) dx. 6