Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
Related questions
Question
Find the volume of the solid that results when the region enclosed by the given curves is evolved about the y-axis for question #25
![Chapter 6 / Applications of the Definite Integral in Geometry, Science, and Engineering
23. x = y, x = y +2
24. x = 1- y?, x = 2 + y?, y = -1, y = 1
6.
AY
y = ycos x
(1, 1)
25. y = In x, x = 0, y = 0, y = 1
y x
1-x2
26. y = Vx2
(x > 0), x = 0, y = 0, y = 2
27-30 True-False Determine whether the statement is true or
false. Explain your answer. [In these exercises, assume that a
solid S of volume V is bounded by two parallel planes perpen-
dicular to the x-axis at x = a and x = b and that for each x in
[a, b), A(x) denotes the cross-sectional area of S perpendicular
to the x-axis.]Y
8.
(2, 3)
y =x 1/
27. If each cross section of S perpendicular to the x-axis is a
square, then S is a rectangular parallelepiped (i.e., is box
shaped).
x = V1 +y
x%3D
28. If each cross section of S is a disk or a washer, then S is a
solid of revolution.
29. If x is in centimeters (cm), then A(x) must be a quadratic
function of x, since units of A(x) will be square centimeters
(cm?).
30. The average value of A(x) on the interval [a, b] is given by
v/(b - a).
Find the volume of the solid whose base is the region
bounded between the curve y = x and the x-axis from
x = 0 to x =2 and whose cross, sections taken perpendic-
ular to the x-axis are squares.
31. Find the volume of the solid that results when the region
above the x-axis and below the ellipse
Find the volume of the solid whose base is the region
bounded between the curve y = sec x and the x-axis from
x = 7/4 to x = 7/3 and whose cross sections taken per-
pendicular to the x-axis are squares.
x²
a?
= 1 (a > 0, b > 0)
b2
is revolved about the x-axis.
32. Let V be the volume of the solid that results when the region
enclosed by y = 1/x, y = 0, x = 2, and x =b (0 < b< 2)
is revolved about the x-axis. Find the value of b for which
V = 3.
33. Find the volume of the solid generated when the region
enclosed by y = Vx + 1, y =
about the x-axis. [Hint: Split the solid into two parts.]
34. Find the volume of the solid generated when the region
-18 Find the volume of the solid that results when the region
closed by the given curves is revolved about the x-axis.
1. y = v25 – x2, y = 3
2. y = 9 - x, y = 0
13. x = Vỹ, x = y/4
V2r, and y =0 is revolved
4. y = sinx, y = cos x, x = 0, x= T/4
[Hint: Use the identity cos 2r = cos? x - sin? x.]
V3D
5. y = e", y = 0, x = 0, x = In 3
6. y = e-2, y = 0, x = 0, x = 1
enclosed by y =
about the x-axis. [Hint: Split the solid into two parts.]
Vx, y = 6 - x, and y = 0 is revolved
17. y =
,x= -2, x = 2, y = 0
V4 +x
3x
FOCUS ON CONCEPTS
VIt eor *= 0, x = 1, y = 0
19. Find the volume of the solid whose base is the region
bounded between the curve y = x' and the y-axis from
y = 0 to y = 1 and whose cross sections taken perpendic-
ular to the y-axis are squares.
18. y =
35. Suppose that f is a continuous function on (a, bl, and
let R be the region between the curve y = f(x) and
the line y = k from x = a to x = b. Using the method
of disks, derive with explanation a formula for the vol-
ume of a solid generated by revolving R about the line
y = k. State and explain additional assumptions, if any,
that you need about f for your formula.
20. Find the volume of the solid whose base is the region en-
closed between the curve x = 1 - y and the y-axis and
whose cross sections taken perpendicular to the y-axis are
36. Suppose that v and w are continuous functions on [c, d],
and let R be the region between the curves x =
x = w(y) from y =c to y = d. Using the method of
washers, derive with explanation a formula for the vol.
ume of a solid generated by revolving R about the line
x= k. State and explain additional assumptions, if any.
that you necd about v and w for your formula.
squares.
= v(y) and
21-26 Find the volume of the solid that results when the region
enclosed by the given curves is revolved about the y-axis,
21. x= csc y, y= T/4, y = 3/4, x = 0
22. y = x, x = y?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9372b6fe-9d83-4cc3-b9b6-3000d02c5781%2Fe9b95b7b-ce1d-4cae-bf72-f004df16a6df%2Fr3tytqe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Chapter 6 / Applications of the Definite Integral in Geometry, Science, and Engineering
23. x = y, x = y +2
24. x = 1- y?, x = 2 + y?, y = -1, y = 1
6.
AY
y = ycos x
(1, 1)
25. y = In x, x = 0, y = 0, y = 1
y x
1-x2
26. y = Vx2
(x > 0), x = 0, y = 0, y = 2
27-30 True-False Determine whether the statement is true or
false. Explain your answer. [In these exercises, assume that a
solid S of volume V is bounded by two parallel planes perpen-
dicular to the x-axis at x = a and x = b and that for each x in
[a, b), A(x) denotes the cross-sectional area of S perpendicular
to the x-axis.]Y
8.
(2, 3)
y =x 1/
27. If each cross section of S perpendicular to the x-axis is a
square, then S is a rectangular parallelepiped (i.e., is box
shaped).
x = V1 +y
x%3D
28. If each cross section of S is a disk or a washer, then S is a
solid of revolution.
29. If x is in centimeters (cm), then A(x) must be a quadratic
function of x, since units of A(x) will be square centimeters
(cm?).
30. The average value of A(x) on the interval [a, b] is given by
v/(b - a).
Find the volume of the solid whose base is the region
bounded between the curve y = x and the x-axis from
x = 0 to x =2 and whose cross, sections taken perpendic-
ular to the x-axis are squares.
31. Find the volume of the solid that results when the region
above the x-axis and below the ellipse
Find the volume of the solid whose base is the region
bounded between the curve y = sec x and the x-axis from
x = 7/4 to x = 7/3 and whose cross sections taken per-
pendicular to the x-axis are squares.
x²
a?
= 1 (a > 0, b > 0)
b2
is revolved about the x-axis.
32. Let V be the volume of the solid that results when the region
enclosed by y = 1/x, y = 0, x = 2, and x =b (0 < b< 2)
is revolved about the x-axis. Find the value of b for which
V = 3.
33. Find the volume of the solid generated when the region
enclosed by y = Vx + 1, y =
about the x-axis. [Hint: Split the solid into two parts.]
34. Find the volume of the solid generated when the region
-18 Find the volume of the solid that results when the region
closed by the given curves is revolved about the x-axis.
1. y = v25 – x2, y = 3
2. y = 9 - x, y = 0
13. x = Vỹ, x = y/4
V2r, and y =0 is revolved
4. y = sinx, y = cos x, x = 0, x= T/4
[Hint: Use the identity cos 2r = cos? x - sin? x.]
V3D
5. y = e", y = 0, x = 0, x = In 3
6. y = e-2, y = 0, x = 0, x = 1
enclosed by y =
about the x-axis. [Hint: Split the solid into two parts.]
Vx, y = 6 - x, and y = 0 is revolved
17. y =
,x= -2, x = 2, y = 0
V4 +x
3x
FOCUS ON CONCEPTS
VIt eor *= 0, x = 1, y = 0
19. Find the volume of the solid whose base is the region
bounded between the curve y = x' and the y-axis from
y = 0 to y = 1 and whose cross sections taken perpendic-
ular to the y-axis are squares.
18. y =
35. Suppose that f is a continuous function on (a, bl, and
let R be the region between the curve y = f(x) and
the line y = k from x = a to x = b. Using the method
of disks, derive with explanation a formula for the vol-
ume of a solid generated by revolving R about the line
y = k. State and explain additional assumptions, if any,
that you need about f for your formula.
20. Find the volume of the solid whose base is the region en-
closed between the curve x = 1 - y and the y-axis and
whose cross sections taken perpendicular to the y-axis are
36. Suppose that v and w are continuous functions on [c, d],
and let R be the region between the curves x =
x = w(y) from y =c to y = d. Using the method of
washers, derive with explanation a formula for the vol.
ume of a solid generated by revolving R about the line
x= k. State and explain additional assumptions, if any.
that you necd about v and w for your formula.
squares.
= v(y) and
21-26 Find the volume of the solid that results when the region
enclosed by the given curves is revolved about the y-axis,
21. x= csc y, y= T/4, y = 3/4, x = 0
22. y = x, x = y?
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