Question 1

The equations y = x³, y = 0, and x = 1 define the bounds of a plane region. Find the volume of the solid obtained by rotating the region about the x-axis.

Accepted Answer

A

Question 2

The equations x = 2, x = 4, y = 1/x, and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

Accepted Answer

B

Question 3

Find the volume of the solid obtained by rotating about the x-axis the region lying under the curve  above the x-axis and to the left of the y-axis.

Accepted Answer

A

Question 4

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x² - 2x.

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 5

The region R is bounded by y = ln x, y = 0, x = 1, and x = 2. Find the volume of the solid obtained by revolving R about the y-axis.

Accepted Answer

The answer of The region R is bounded by y...

Question 6

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the x-axis.

Accepted Answer

The answer of Find the volume of a solid generated...

Question 7

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the y-axis.

Accepted Answer

The answer of Find the volume of a solid generated...

Question 8

The equations x = -1, x = 0, y =  , and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

Accepted Answer

The answer of The equations x = -1, x =...

Question 9

If R is the region enclosed by the graphs of y = f(x) and y = g(x) from x = a to x = b (as shown in the figure below), then the volume V of the solid generated by revolving the region R about the line y = -2 is V = $\pi$  dx.

Accepted Answer

The answer of If R is the region enclosed by...

Question 10

The equations y² = 4x and x² = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y² = 4x and above the curve x² = 4y about (a) the x-axis and (b) the y-axis.

Accepted Answer

The answer of The equations y² = 4x and x²...

Question 11

Find the volumes of solids generated when the ellipse  +  = 1 (where a > 0 and b > 0) is rotated about (a) the x-axis and (b) the y-axis.

Accepted Answer

The answer of Find the volumes of solids generated when...

Question 12

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the x-axis.

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 13

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the y-axis.

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 14

The region R is the portion of the first quadrant that is below the parabola y² = 8x and above the hyperbola y² - x² = 15. Find the volume of the solid obtained by revolving R about the x-axis.

Accepted Answer

The answer of The region R is the portion of...

Question 15

Find the volume of the solid generated by revolving the triangular region bounded by the lines y = x, y = -x, and x = a (where a > 0) about its edge x = a.

Accepted Answer

The answer of Find the volume of the solid generated...

Question 16

Find the volume of the solid generated when the region lying under the curve y = 4 - x² and above the x-axis is rotated about the line y = -1.

Accepted Answer

The answer of Find the volume of the solid generated...

Question 17

A cylindrical hole of radius r cm is drilled through the centre of a ball of radius R cm (where R > r). Find the volume of the remaining part of the ball.

Accepted Answer

The answer of A cylindrical hole of radius r cm...

Question 18

Find the volume of a solid formed by revolving the disk bounded by a circle of radius a cm about a line tangent to that circle.

Accepted Answer

The answer of Find the volume of a solid formed...

Question 19

The plane region bounded by the curve  +  = 1 is revolved about the line x = 2. Find the volume of the solid generated.

Accepted Answer

The answer of The plane region bounded by the curve...

Question 20

For what real values of the constant k does the region lying under the curve y =  above the x-axis and to the right of the line x = 1 have infinite area but gives rise to a solid with finite volume when rotated about the x-axis?

Accepted Answer

The answer of For what real values of the constant...

Quiz 8: Applications of Integration

The equations y = x³, y = 0, and x = 1 define the bounds of a plane region. Find the volume of the solid obtained by rotating the region about the x-axis.

The equations x = 2, x = 4, y = 1/x, and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x² - 2x.

The region R is bounded by y = ln x, y = 0, x = 1, and x = 2. Find the volume of the solid obtained by revolving R about the y-axis.

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the x-axis.

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the y-axis.

The equations x = -1, x = 0, y = , and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

The equations y² = 4x and x² = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y² = 4x and above the curve x² = 4y about (a) the x-axis and (b) the y-axis.

Find the volumes of solids generated when the ellipse + = 1 (where a > 0 and b > 0) is rotated about (a) the x-axis and (b) the y-axis.

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the x-axis.

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the y-axis.

The region R is the portion of the first quadrant that is below the parabola y² = 8x and above the hyperbola y² - x² = 15. Find the volume of the solid obtained by revolving R about the x-axis.

Find the volume of the solid generated by revolving the triangular region bounded by the lines y = x, y = -x, and x = a (where a > 0) about its edge x = a.

Find the volume of the solid generated when the region lying under the curve y = 4 - x² and above the x-axis is rotated about the line y = -1.

A cylindrical hole of radius r cm is drilled through the centre of a ball of radius R cm (where R > r) . Find the volume of the remaining part of the ball.

Find the volume of a solid formed by revolving the disk bounded by a circle of radius a cm about a line tangent to that circle.

The plane region bounded by the curve + = 1 is revolved about the line x = 2. Find the volume of the solid generated.

Quiz 8: Applications of Integration

The equations y = x3, y = 0, and x = 1 define the bounds of a plane region. Find the volume of the solid obtained by rotating the region about the x-axis.

The equations x = 2, x = 4, y = 1/x, and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

Find the volume of the solid obtained by rotating about the x-axis the region lying under the curve above the x-axis and to the left of the y-axis.

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x2 - 2x.

The region R is bounded by y = ln x, y = 0, x = 1, and x = 2. Find the volume of the solid obtained by revolving R about the y-axis.

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = π\piπ is rotated about the x-axis.

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = π\piπ is rotated about the y-axis.

The equations x = -1, x = 0, y = , and y = 0 define the bounds of a region of the plane. Find the volume of the solid obtained by rotating the region about the x-axis.

If R is the region enclosed by the graphs of y = f(x) and y = g(x) from x = a to x = b (as shown in the figure below), then the volume V of the solid generated by revolving the region R about the line y = -2 is V = π\piπ dx.

The equations y2 = 4x and x2 = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y2 = 4x and above the curve x2 = 4y about (a) the x-axis and (b) the y-axis.

Find the volumes of solids generated when the ellipse + = 1 (where a > 0 and b > 0) is rotated about (a) the x-axis and (b) the y-axis.

Find the volume of the solid obtained by rotating the region inside the circle x2 + y2 = 6 and above the parabola y = x2 about the x-axis.

Find the volume of the solid obtained by rotating the region inside the circle x2 + y2 = 6 and above the parabola y = x2 about the y-axis.

The region R is the portion of the first quadrant that is below the parabola y2 = 8x and above the hyperbola y2 - x2 = 15. Find the volume of the solid obtained by revolving R about the x-axis.

Find the volume of the solid generated by revolving the triangular region bounded by the lines y = x, y = -x, and x = a (where a > 0) about its edge x = a.

Find the volume of the solid generated when the region lying under the curve y = 4 - x2 and above the x-axis is rotated about the line y = -1.

A cylindrical hole of radius r cm is drilled through the centre of a ball of radius R cm (where R > r) . Find the volume of the remaining part of the ball.

Find the volume of a solid formed by revolving the disk bounded by a circle of radius a cm about a line tangent to that circle.

The plane region bounded by the curve + = 1 is revolved about the line x = 2. Find the volume of the solid generated.

For what real values of the constant k does the region lying under the curve y = above the x-axis and to the right of the line x = 1 have infinite area but gives rise to a solid with finite volume when rotated about the x-axis?

The equations y = x³, y = 0, and x = 1 define the bounds of a plane region. Find the volume of the solid obtained by rotating the region about the x-axis.

Find the volume of the solid obtained by rotating about the x-axis the plane region lying under the x-axis and above the curve y = x² - 2x.

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the x-axis.

Find the volume of a solid generated when the region under the curve y = sin x and above the x-axis from x = 0 to x = $\pi$ is rotated about the y-axis.

The equations y² = 4x and x² = 4y define the bounds of a plane region. Find the volume of the solid obtained by rotating the region to the right of the curve y² = 4x and above the curve x² = 4y about (a) the x-axis and (b) the y-axis.

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the x-axis.

Find the volume of the solid obtained by rotating the region inside the circle x² + y² = 6 and above the parabola y = x² about the y-axis.

The region R is the portion of the first quadrant that is below the parabola y² = 8x and above the hyperbola y² - x² = 15. Find the volume of the solid obtained by revolving R about the x-axis.

Find the volume of the solid generated when the region lying under the curve y = 4 - x² and above the x-axis is rotated about the line y = -1.