AP Calculus Testbank (Chapter 7) (Mr. Surowski)

Transcription

1 AP Calculus Testbank (Chapter 7) (Mr. Surowski) Part I. Multiple-Choice Questions. Suppose that a function = f() is given with f() for 4. If the area bounded b the curves = f(), =, =, and = 4 is revolved about the -ais, then the volume of the resulting solid would best be computed b the method of (A) disks/washers (B) shells (C) known cross sections.. Suppose that a function = f() is given with f() for 4. If the area bounded b the curves = f(), =, =, and = 4 is revolved about the -ais, then the volume of the solid of revolution is given b (A) π (B) π (C) π (D) π (E) π f() d f() d + f() d f() d f() d

2 3. A parabola is drawn having focus (, ) and directri = 4. The definite integral representing the arc length of that portion of the parabola on or above the -ais is given b (A) 4 d (B) 3 4 d = 4 (C) (D) (E) d 4 + d d 4 + F (, ) 4. Consider the solid of revolution formed b revolving the area bounded b the curve = /, the -ais, the line = and the line = a, (a > ) about the -ais. The integral representing the volume of this solid is (A) π (B) π (C) π (D) π (E) π d d d d d = = a

3 5. Consider the surface of revolution formed b revolving the the curve = /, a about the -ais. Then the surface area is given b the definite integral (A) π (B) π (C) π (D) π (E) π d d + 4 d 3 ( + ) + 4 d d 6. Which of the following integrals correctl gives the area of the region consisting of all points above the -ais and below the curve = 8 +? (A) (B) (C) (D) (E) 4 4 ( 8) d (8 + ) d (8 + ) d ( 8) d (8 + ) d.

4 7. A solid is generated with the region in the first quadrant bounded b the graph of = + sin, the line = π, the -ais, and the -ais is revolved about the -ais. Its volume is found b evaluating which of the following integrals? (A) π (B) π (C) π (D) π (E) π π π π ( + sin 4 ) d ( + sin ) d ( + sin 4 ) d ( + sin ) d ( + sin ) d. 8. The volume generated b revolving about the -ais the region above the curve = 3, below the line =, and between = and = is (A) π 4 (B).43π (C) π 7 (D).643π (E) 6π 7 9. Find the distance traveled (to three decimal places) from t = to t = 5 seconds, for a particle whose velocit is given b v(t) = t + ln t. (A) 6. (B).69 (C) 6.47 (D).8 (E) Find the area of the region bounded b the parabolas = and = 6 (A) 9 (B) 7 (C) 6 (D) 9 (E) 8

5 . What is the area of the region in the first quadrant enclosed b the graph of = e 4 and the line =.5? (A).4 (B).56 (C).48 (D).3 (E).349. The base of a solid S is the region enclosed b the graph of =, the -ais, and the -ais. If the cross-sections of S perpendicular to the -ais are semicircles, then the volume of S is (A) 5π 3 (B) π 3 (C) 5π 3 (D) 5π 3 (E) 45π 3 3. The volume of the solid that results when the area between the curve = e and the line =, from = to =, is revolved around the -ais is (A) π(e 4 e ) (B) π (e4 e ) (C) π (e e) (D) π(e e) (E) πe 4. What is the volume of the solid generated b rotating about the -ais the region enclosed b = sin and the -ais, from = to = π? (A) π (B) π (C) 4π (D) (E) 4

6 Part II. Free-Response Questions. Given the velocit function v(t) = t 4 + t, t, () =, (a) determine the terminal position of the particle, and (b) determine the total distance traveled b the particle. The terminal position is given b () = () + v(t) dt = ( (t ) dt = 4 + t ln(4 + t ) ) tan (/) = ln π 8. The total distance traveled b the particle is given b distance = = = = = ( speed dt v(t) dt t dt 4 + 4t ( t) dt t (t ) dt 4 + t tan (/) ) ln(4 + t ) + = π 8 ln(5/4) + ln(8/5) π 8 + tan (/) = ln(3/5) + tan (/) ( ln(4 + t ) ) tan (/). Given the velocit function v(t) = + sin t, t π/6, with () =, (a) determine the terminal position of the particle, and (b) determine the total distance travelled b the particle. (a) The terminal position is (π/6) = ()+ π/6 (+ sin t) dt = (t cos t) π/6 = π

7 (b) Since v(t) on t π/6, we see that velocit and speed are the same on this interval. Therefore total distance = π/6 ( + sin t) dt = π Given the velocit function v(t) = t cos πt, t.5, with () =, (a) determine the terminal position of the particle, and (b) determine the total distance travelled b the particle. (a) The terminal position is (.5) = () +.5 t cos πt dt = =.5 π π. ( t π sin πt + ).5 cos πt π (b) Since v(t) on the interval π t 3π, we have total distance = =.5 speed dt = ( t π sin πt + ).5 cos πt π.5 = π π + 3 π + π = 5 π π.5 t cos πt dt t cos πt dt.5 ( t π sin πt + ).5 cos πt π.5 4. The velocit function of a particle has the graph depicted below. Find the total distance travelled b the particle over the first five seconds. v (cm/sec) v = v(t) t (sec) The total distance traveled is just the total area under the velocit graph. Using simple geometr one discovers that the total distance is cm.

8 5. Suppose that a particle is initiall at rest at the origin, but at time t = a force is applied to the particle which results in an acceleration of + cm/sec. Locate this particle on the -ais after 5 seconds. This is simple: after two simple integrations one has (t) = 5t, so the particle occupies position = 75 cm after 5 seconds. 6. Water is flowing from a faucet into a one-litre bottle at a rate of r(t) = te t l/min. After minutes the water is turned up to a constant rate of of. l/min. (a) Graph the function r = r(t) depicting the rate of flow of water..3 r (l/min).. t (min) 3 4 Equation : =e^( )( )/( ) Equation : =(.)( )/( ) (b) Will the bottle be full after 4 minutes? The total amount of water flowing into the bottle over the first four minutes is ( A = r(t) dt = te t dt+ (.) dt = te t ) 4 e t.4 = e 4 4 e l. Therefore, the bottle will 4 not be full. (Alternativel, ou could have just computed te t dt numericall on our calculator, without having to resort to integration b parts.) +

9 7. Suppose that a particle is resting at the origin and that a force of F = F (t) cm/sec, t, is applied to the particle over the interval t <. Assuming that F (t) > over this interval, compute lim (t) and justif our answer. Since a positive t force is applied, the particle will eperience positive acceleration. This will move the particle off to infinit as t. That is to sa, lim (t) = + t 8. Graph the region bounded b the curves = and + = 3 and compute its area. The points of intersection occur where = ±. Therefore the area between the curves is given b the integration along the -ais (note how smmetr is being used): Area = [(3 ) ] d = (3 3 ) = Graph the region bounded b the curves Equation = : +^=3 + 3 and = 3 5, ( ) and compute itsequation area. 3: =^ Note that the points of intersection of these curves are = and = ±. As we re onl interested in, the area involved is Area = [( +3) ( 3 5)] d = Equation : = ²+3 Equation 3: =³ ² 5 ( 3 +8) d = 4 +4 = 8.

10 . Set up an integral (without evaluating it) that will compute the area of the region 9 + 4,. Whether we integrate along the - or -ais is immaterial. We ll set up both, making full use of smmetr: Area = d = 6 4 d. (Note that the common value is 6π.) Equation : ²/9+^/4

11 . Consider the region bounded b the curve = / p, =, = a, and the -ais. (a) Compute the volume of the solid of revolution obtained b revolving the above region about the -ais. (b) If V (a) represents the volume given in part (a) above, compute lim V (a). a (c) There is a value p such that if p p, the limit in part (b) above is infinite and if p > p, the limit in part (b) above is finite. Find this number p.. Consider the region bounded b the curve = / p, =, = a, and the -ais. (a) Compute the volume of the solid of revolution obtained b revolving the above region about the -ais. (b) If V (a) represents the volume given in part (a) above, compute lim V (a). a (c) There is a value p such that if p p, the limit in part (b) above is infinite and if p > p, the limit in part (b) above is finite. Find this number p.

12 3. Consider the surface generated b revolving the curve = / p, a, about the -ais. (a) Epress the area of the above surface as an integral (ou probabl won t be able to evaluate this integral). (b) If S(a) represents the area given in part (a) above, compute lim a S(a). (c) There is a value p such that if p p, the limit in part (b) above is infinite and if p > p, the limit in part (b) above is finite. Find this number p. 4. The region below is revolved about the -ais to form a solid of revolution. Find the volume of this solid. 4 = /4 - / = / 5. A solid object has a flat base formed b the region enclosed b the parabola with focus having coordinates (, ) and directri = 4 and b the -ais. Each cross section is an equilateral triangle perpendicular to the base and parallel to the directri. Compute the volume of this object.

13 6. Compute the length of that section of the curve = 4 /4 + /8 that joins (3/8, ) to the point (9/3, ). 7. Consider the solid of revolution formed b revolving the area bounded b the curve = /, the -ais, the line = and the line = a about the -ais. Let V (a) represent this volume and compute lim a V (a). 8. If S(a) represents the surface of revolution of problem 5, compute lim a S(a). 9. Use integral calculus to show that the volume of a right circular cone of height h and base area A is 3 Ah.. Suppose that a metal chain weighing newtons/m is hanging over a building. Assuming that the building is 3 m tall, and that the chain is just touching the ground, what is the total work required to pull the chain onto the top of the building?. Suppose that an object rests at the point = on the -ais. We then start pushing this bo in the positive direction, giving the bo a speed of e t/ m/sec. Assume that there is a force due to friction, the magnitude of which is / the speed of the bo. Find the total work needed to push the bo for seconds.. Suppose that a large clindrical drum of height meters and radius 3 meters is full of a fluid whose weight densit is, N/m 3. Find the total force on the side of the clindrical drum. (Recall: the fluid pressure at a depth h is p = wh, where w is the weight-densit in this case, N/m 3.)

14 3. Suppose that we have a lake full of fish the weights of which are modeled b a normal distribution with mean.77 kg and standard deviation of. kg. Epress the probabilit as an integral, written as eplicitel as possible that a randoml-selected fish will have its weight somewhere between.5 kg and. kg. What is the probabilit that a randoml selected fish will have its weight somewhere between.65 kg and.89 kg? 4. Assume that there is a heav bo sitting outside on the pavement. We are going to move this bo a total of feet b sliding it along the pavement. The relevant force here is that of friction, which we shall assume is proportional to the speed at which we slide the bo. Which will result in less work, sliding the bo quickl over the necessar feet or sliding it slowl? Please eplain.