Practice Exercises

Area

In Questions A1–A11, choose the alternative that gives the area of the region whose boundaries are given.

A1. The curve y = x², y = 0, x = −1, and x = 2.

A2. The parabola y = x² − 3 and the line y = 1.

A3. The curve x = y² − 1 and the y-axis.

A4. 

The parabola y² = x and the line x + y = 2.

A5. The curve , the x-axis, and the vertical lines x = −2 and x = 2.

A6. 

The parabolas x = y² − 5y and x = 3y − y².

A7. The curve  and x + y = 3.

A8. In the first quadrant, bounded below by the x-axis and above by the curves of y = sin x and y = cos x.

A9. Bounded above by the curve y = sin x and below by y = cos x from  to .

A10. The curve y = cot x, the line , and the x-axis.

A11. The curve y = x³ − 2x² − 3x and the x-axis.

A12.

 

The total area bounded by the cubic x = y³ − y and the line x = 3y is equal to

(A) 4
(B) 
(C) 8
(D) 16

A13. The area bounded by y = e^x, y = 2, and the y-axis is equal to

(A) 3 − e
(B) e² + 1
(C) 2 ln 2 − 1
(D) 2 ln 2 − 3

CHALLENGE A14. The area enclosed by the curve y² = x(1 − x) is given by

A15. The figure below shows part of the curve y = x³ and a rectangle with two vertices at (0,0) and (c,0). What is the ratio of the area of the rectangle to the shaded part of it above the cubic?

(A) 3 : 4
(B) 5 : 4
(C) 4 : 3
(D) 3 : 1

Volume

In Questions A16–A22, the region whose boundaries are given is rotated about the line indicated. Choose the alternative that gives the volume of the solid generated.

A16. y = x², x = 2, and y = 0; about the x-axis.

A17. y = x², x = 2, and y = 0; about the y-axis.

A18. The first-quadrant region bounded by y = x², the y-axis, and y = 4; about the y-axis.

A19. y = x² and y = 4; about the x-axis.

A20. y = x² and y = 4; about the line y = 4.

CHALLENGE A21. An arch of y = sin x and the x-axis; about the x-axis.

A22. A trapezoid with vertices at (2,0), (2,2), (4,0), and (4,4); about the x-axis.

A23. The base of a solid is a circle of radius a and is centered at the origin. Each cross section perpendicular to the x-axis is a square. The solid has volume

A24. The base of a solid is the region bounded by the parabola x² = 8y and the line y = 4, and each plane section perpendicular to the y-axis is an equilateral triangle. The volume of the solid is

A25. The base of a solid is the region bounded by y = e^−x, the x-axis, the y-axis, and the line x = 1. Each cross section perpendicular to the x-axis is a square. The volume of the solid is

*Length of Curve (Arc Length)

CHALLENGE *A26.

The length of the curve y² = x³ cut off by the line x = 4 is

*A27. The length of the curve y = ln (cos x) from  to  equals

Improper Integrals

*A28. 

(A) 1
(B) 
(C) −1
(D) divergent

*A29. 

(A) 1
(B) 
(C) 
(D) divergent

*A30. 

*A31. 

(A) 6
(B) 0
(C) 
(D) divergent

*A32. 

(A) 2
(B) −2
(C) 0
(D) divergent

*A33. 

(A) −2
(B) 
(C) 2
(D) divergent

*A34. Find the area in the first quadrant under the curve y = e^−x.

(A) 1
(B) e
(C) 
(D) divergent

*A35. 

Find the area in the first quadrant under the curve .

*A36. Find the area in the first quadrant above y = 1, between the y-axis and the curve .

(A) 1
(B) 2
(C) 
(D) divergent

*A37.

Find the area between the curve  and the x-axis.

(A) 2π
(B) 4π
(C) 8π
(D) divergent

*A38.

Find the area above the x-axis, between the curve  and its asymptotes.

(A) π
(B) 2π
(C) 4π
(D) divergent

*A39. Find the volume of the solid generated when the region bounded above by , at the left by x = 1, and below by y = 0 is rotated about the x-axis.

(A) 
(B) π
(C) 2π
(D) divergent

*A40. Find the volume of the solid generated when the first-quadrant region under y = e^−x is rotated about the x-axis.

(A) 
(B) π
(C) 2π
(D) divergent

Area

In Questions B1–B4, choose the alternative that gives the area of the region whose boundaries are given.

B1.The area bounded by the parabola y = 2 − x² and the line y = x − 4 is given by

B2. Suppose the following is a table of coordinates for y = f(x), given that f is continuous on [1,8]

If a trapezoidal sum in used, with n = 4, then the approximate area under the curve, from x = 1 to x = 8, to two decimal places, is

(A) 24.87
(B) 39.57
(C) 49.74
(D) 59.91

*B3. The area A enclosed by the four-leaved rose r = cos 2θ equals, to three decimal places,

(A) 0.785
(B) 1.571
(C) 3.142
(D) 6.283

*B4. The area bounded by the small loop of the limaçon r = 1 − 2 sin θ is given by the definite integral

Volume

In Questions B5–B10, the region whose boundaries are given is rotated about the line indicated. Choose the alternative that gives the volume of the solid generated.

B5. y = x² and y = 4; about the line y = −1.

B6. y = 3x − x² and y = 0; about the x-axis.

B7. y = 3x − x² and y = x; about the x-axis.

B8. y = ln x, y = 0, x = e; about the line x = e.

CHALLENGE B9. A sphere of radius r is divided into two parts by a plane at distance h (0 < h < r) from the center. The volume of the smaller part equals

B10. If the curves of f(x) and g (x) intersect for x = a and x = b and if f(x) > g (x) > 0 for all x on (a,b), then the volume obtained when the region bounded by the curves is rotated about the x-axis is equal to

Length of Curve (Arc Length)

*B11. The length of one arch of the cycloid  equals

*B12. The length of the curve of the parabola 4x = y² cut off by the line x = 2 is given by the integral

*B13. The length of x = e^t cos t, y = e^t sin t from t = 2 to t = 3 is equal to

(A) 17.956
(B) 25.393
(C) 26.413
(D) 37.354

Improper Integrals

*B14. Which one of the following is an improper integral?

*B15. Which one of the following improper integrals diverges?

*B16. Which one of the following improper integrals diverges?