MATH 104 Sample problems for first exam - Fall MATH 104 First Midterm Exam - Fall (d) 256 3

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1 MATH 14 Sample problems for first exam - Fall 1 MATH 14 First Midterm Exam - Fall 1. Find the area between the graphs of y = 9 x and y = x + 1. (a) 4 (b) (c) (d) 5 (e) 4 (f) 81. A solid has as its base the region in the xy-plane the region between the curve y = 1 x and the x-axis. The cross-sections of the solid perpendicular to the y-axis 4 are squares with one side in the base. What is the volume of the solid? (a) 7 (b) 144 (c) 18 (d) (e) 4 (f) 8. Let V (b) be the volume of the solid obtained by rotating the area between the graph of y = e 4x and the x-axis for < x < b around the y-axis. What is lim b V (b)? (a) π (b) π (c) π (d) π 4 (e) π (f) π 8 4. Find the length of the part of the curve y = e x e x for x ln. (a) 19 1 (b) 11 1 (c) 8 (d) 9 8 (e) 9 1 (f) e4 1 5 ln(5x) x dx (a) 5 (b) 14 (c) (d) 7 (e) 8 (f) 18

2 . An object moves in such a way that its acceleration at time t seconds is (1 + t) 4 meters per second. If the initial velocity of the object is 5 meters per second, what is the limit of its velocity as t? (a) 7 m/s (b) 8 m/s (c) 9 m/s (d) 1 m/s (e) 11 m/s (f) 1 m/s 7. Find the surface area obtained by rotating the part of the curve y = x 4 for x around the y-axis. ( ) (a) 5 π ( ) (d) 1 π 9 ( 1 (b) 5 ( 8 1 (e) ) π ) π ( 1 8 (c) ( (f) Find the x-coordinate of the centroid of the quarter-disk in the first quadrant bounded by the coordinate axis and the graph of y = x ) π ) π (a) 4 π (b) π (c) 8 π (d) 4 π (e) π (f) 8 π 9. The region between the graph of y = 1 x and the x-axis is rotated around the line x =. What is the volume of the resulting solid? (a) π (b) 4π (c) 8π (d) 1π (e) π (f) 1π 1. When the region between the graph of y = f(x), the x-axis and the line x = b for b > is rotated around the x-axis, the volume of the resulting solid is πb. What is f(x)? (a) 4x 5/ (b) x / (c) x 5/ (d) x 7/ (e) x / (f) 4x 7/

3 MATH 14 First Midterm Exam - Fall Find the area between the graphs of y = x + x + 5 and y = x x. (a) 4 (b) (c) 4 (d) (e) (f) 81. A solid has as its base the region in the xy-plane the region between the curve y = 1 x4 and the x-axis. The cross-sections of the solid perpendicular to the x-axis 81 are semicircles with the flat side (i.e., the diameter) in the base. What is the volume of the solid? (a) 8π (b) π (c) π 5 (d) 1π 45 (e) 8π 45 (f) 4π. Let V (a) be the volume of the solid obtained by rotating the area between the graph of y = 1 and the x-axis for 1 < x < a around the y-axis. x What is lim a V (a)? (a) 4π (b) π (c) π (d) π (e) π (f) π 4. Find the length of the part of the curve y = 1 1 x + 1 x for 1 x. (a) 17 1 (b) 1 1 (c) 19 1 (d) 1 (e) 9 1 (f) π/ cos(x) 1 + sin (x) dx (a) π (b) π 4 (c) π 8 (d) π 1 (e) π 1 (f) π 4

4 . An object moves in such a way that its acceleration is e t/ meters per second. If the initial velocity of the object is 5 meters per second, what is the limit of its velocity as t? (a) 9 m/s (b) 1 m/s (c) 14 m/s (d) 1 m/s (e) 17 m/s (f) m/s 7. Find the surface area obtained by rotating the part of the curve y = x + 1 for x 1 around the y-axis. (a) π (5 5 1) (b) π ( ) (c) π (7 7 1) (d) π (5 5 1) (e) π (8 8 1) (f) π ( ) 8. Find the volume obtained by rotating the region between the curve y = 1 4x and the x-axis around the x-axis. (a) π 9 (b) π 4 (c) π (d) 4π 9 (e) π (f) π 9. The region inside the triangle with vertices (1, ), (1, 1) and (, ) is rotated around the line x = 5. What is the volume of the resulting solid? (a) π (b) 5π (c) 8π (d) 11π (e) 14π (f) 17π 1. The region between the x-axis, the y-axis, and the line y = 8 8x is revolved around the y-axis to generate a cone. The centroid (center of mass assuming constant density) of this cone is on the y-axis. What is its y-coordinate? (Hint: Use disks. What is the center of mass of a disk? What fraction of the total mass is in each disk? It will save an integral to know that the volume of a cone is 1 πr h, where r is the radius of the base of the cone and h is its height.) (a) 8 (b) 1 (c) (d) 1 (e) (f) 4

5 Forty more practice problems 1. e 4 1 e x ln x dx = (a) (b) (c) 4 (d) e (e) e 4 e. We know that the graph of a function ϕ(x) passes through the point (4,). We also know that 1 ϕ (x) for all x in the interval [, 1]. What are the minimum and maximum values possible values for ϕ(7)? (a) min = 7, max = 1 (b) min = 4, max = 7 (c) min = 1, max = 4 (d) min =, max = 1 (e) min = 4, max = 1 (f) min =, max =. Find the area between the graphs of y = 4x + 5 and y = x. (a) 4 (b) (c) 4 (d) (e) (f) What is the volume of the solid obtained by rotating the part of the graph of y = cos x sin x between x = and x = π around the x-axis? (a) π (b) π (c) π (d) π 4 (e) π b arctan x 5. lim b 1 + x dx = (a) π (b) π (c) π 4 (d) π 8 (e) +. The area between a curve y = f(x) and the x-axis between x = 1 and x = b is rotated around the x axis and the volume of the resulting solid is π (ln b), for all b > 1. What is f(x)? (a) (ln x) (b) ln x x (c) ln x x (d) (ln x) x (e) (ln x) x

6 7. π sin (x) dx = (a) (b) 1 (c) 1/ (d) π/ (e) π (f) π 8. Suppose that f(x) and g(x) are two differentiable functions, and we know the following values: Then 1 f() =, f(1) =, g() = 1, g(1) = 4. ( f(x) g (x) + f(x)f (x)g(x) ) dx = (a) (b) 1 (c) 14 (d) 1 (e) (f) The area bounded by the curves y = x 1 and y = x + 7 is (a) 4 (b) 9 (c) 1 (d) 4 (e) (f) f(x) 1. If f(x) is continuous for all x and lim x x 11. = 4, then (a) f() = 1 and f () = 4 (b) f() = and f () = 4 (c) f() = 4 and f () = (d) f() = and f () = 1 (e) f() = and f () = (f) f() = 4 and f () = 1 d sin x e t dt = dx (a) e sin x (b) cos x e sin x (c) sin x cos x e sin x (d) cos x + e sin x (e) sin x e sin x + e sin x (f) e cos x 1. e 1/e (ln x) dx = x (a) e e (b) e e (d) (e) e 1 e (c) e (f) e 1. What is the average value of the function f(x) = (a) π (b) π/ 54 (c) π 18 arcsin x 1 x on the interval [, 1 ]? (d) π/ 7 (e) π 48

7 14. What is the volume of the solid obtained by rotating the region bounded by the graphs of y = x, y = x and y = around the x-axis? (a) 5π (b) π (c) π (d) 11π (e) 5π (f) 7π 5. The functions cosh x and sinh x are defined as cosh x = 1 (ex + e x ) and sinh x = 1 (ex e x ). Calculate the area bounded by the curves y = cosh x, y = sinh x, x = and x = b. What is the limit of this area as b? (a) 1 (b) e (c) e + 1 e (d) 1 e (e) 1. Suppose f(x) is a monotonically increasing function on the interval [, ), and that f() = and x lim f(x) =. Suppose A(b) is the average value of f(x) on the interval [, b] for b >. What is lim A(b)? b (a) (b) 4 (c) 5 (d) (e) (f) 17. What is the volume obtained by revolving the region bounded by y = x 4 and y = 4 x around the line x =? (a) 1π (b) 4π (c) 5π (d) 51π (e) 14π 18. π /4 cos x x dx = (a) (b) 1 (c) (d) (e) (f) 19. Compute (sin x + 8) 7 sin x cos x dx. (a) 1 1 (sin x + 8) 8 + C (b) (cos x + 8) 7 + C (c) 1 (sin x + 8) 8 + C (d) 1 8 (sin (x + 8) 8 + C (e) 1 8 (cos x + 8) 8 + C (f) 1 (sin x + 8) 7 + C. Suppose f is a continuous function and 9 1 f(x) dx =. Then 1 xf(x ) dx = (a) (b) 4 (c) (d) (e) 1 (f)

8 1. 1 e arctan(x) 1 + x dx = (a) e π/4 1 (b) e π/4 (c) e 1 (d) π 4 (e) π 4 e (f) eπ/ e π/4. Find the area between the graphs of y = sin x and y = cos x for x π/4 (a) (b) (c) 1 (d) + 1 (e) 1 (f) 1. π/ sec x tan x dx (a) 1 (b) 1 (c) 11 (d) 7 (e) 1 (f) 1 4. The base of a solid is the triangle in the xy-plane with vertices (,), (1,) and (,1). Cross-sections of the solid perpendicular to the x-axis are squares. What is the volume of the solid? (a) (b) 1 4 (c) 4 (d) 4 (e) 5 4 (f) 1 5. Find the volume of the solid obtained by rotating the area between the graphs of y = x and x = y around the y-axis. (a) π (b) π (c) π (d) π π (e) (f) π A solid has as its base the region in the xy-plane the region between the curve y = 1 x and the x-axis. The cross-sections of the solid perpendicular to the 4 x-axis are squares with one side in the base. What is the volume of the solid? (a) 4 (b) 1 (c) 1 (d) 18 (e) 4 (f) Find the average value of the function f(x) = t 1 + t on the interval x. (a) (b) (c) (d) 1 (e) (f)

9 8. Find the volume obtained by rotating the region between the x-axis, the y-axis and the line x + y = 1 around the line x =. (a) 7π (b) 7π (c) 19π (d) 19π (e) 19π 1 (f) 7π 1 9. Find the length of the part of the curve y = 1 4 x/ 4 x1/ for x 1. (a) 7 (b) 4 (c) 17 1 (d) 19 1 (e) 1 (f) 1. Find the volume of the solid obtained by rotating the region between the graph of y = sin(x/) and the x axis for x π around the x-axis. (a) π 8 (b) π 4 (c) π (d) π (e) π (f) 4π 1. Find the area between the graphs of y = 1 and y = x 4. (a) 8 (b) 1 (c) 1 (d) (e) 8 5 (f) 1. Find the volume of the solid obtained by rotating the area between the graphs of y = x x and y = around the x-axis. (a) π (b) π (c) 4π (d) π (e) π π (f) x 1 + x 8 dx (a) π 4 (b) π 8 (c) π 1 (d) π 1 (e) π (f) π 4 4. Find the volume obtained by rotating the square with corners at the points (, ), (, 1), (1, 1) and (1, ) around the line x = 4. (a) π (b) 5π (c) 7π (d) 9π (e) 11π (f) 1π 5. Let V (b) be the volume obtained by rotating the area between the x-axis and the graph of y = 1 from x = 1 to x = b around the x-axis. What is lim x V (b)? b (a) π 5 (b) π 4 (c) π (d) π (e) π (f)

10 . Let V (a) be the volume obtained by rotating the area between the x-axis and the graph of y = 1 from x = a to x = 1 around the y-axis. What is lim x V (a)? / a + (a) π 5 (b) 4π (c) π (d) 4π (e) π 5 (f) 7. Find the surface area obtained by rotating the part of the curve y = 4 x for x around the y-axis. (a) π (5 5 1) (b) π ( ) (c) π (7 7 1) (d) π (5 5 1) (e) π ( ) (f) π ( ) 8. π x sin (x ) dx (a) π (b) π (c) π (d) π (e) π 4 (f) π 4 9. Find the area between the graphs of y = x 4 + 4x and y = 4x. (a) 1 (b) 8 (c) 1 (d) 7 (e) (f) The average value of the function f(x) on the interval [, b] is b. What is f(x)? (a) (1 + x)e x e x (b) 1 + x (c) cos x + x sin x (d) sin x + x cos x (e) x (f) x 41. Calculate the length: (a) of the part of y = 1 (x + ) / from x = to x = 1. (Answer: 4/) (b) of the part of y = x / from x = 1 to x = 8 (careful!) (Answer: ) 7 (c) of the part of y = ln(cos x) for x π/4. (Answer: ln( + 1)) (d) of the part of y = ln x for 1 x (Answer: ln( 5 1) 1 ln( 1) 1 ln( 5 + 1) + 1 ln( + 1))

11 4. Calculate the surface area obtained by rotating: (a) y = x around the x axis for x 4 (Answer: π ( )) (b) y = x π around the x axis for 1 x (Answer: 7 ( )) (c) y = 9 x around the y axis for 1 x. (Answer: 1π ) (d) y = x π around the x axis for x 1. (Answer: (18 5 ln( 5 + )))