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Question 1

Find the volume when the area bounded by the line y=2x and the curve y=6x-x^2 are rotated about the x axis.

Studyforge Question ID: 607

Question 2

Find the volume when the area bounded by the lines y=2-3x, x=0, and y=0 are rotated about the x--axis.

Studyforge Question ID: 608

Question 3

Find the volume of revolution when the area bounded by the curve y=2-3x, x=0,\text{ and }y=0 is rotated about the y-axis. 

Studyforge Question ID: 609

Question 4

Find the volume of a sphere with a radius of 3 units using the volume by discs method.

Studyforge Question ID: 610

Question 5

(Try this!): Find the volume of a solid built on a base which is bounded by the curves y=\sqrt{x-1} and the x axis from 1 to 5, if the cross sections perpendicular to the x axis are squares.

Studyforge Question ID: 611

Question 6

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the x axis.

y=x^{1/3}+1,\ y=1 from x=0 to x=8

Studyforge Question ID: 612

Question 7

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the x axis.

y=4-x^2,\hspace{5mm}y=0

Studyforge Question ID: 613

Question 8

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the x axis.

y=\sqrt{\sin x},\ \ x=2y

Studyforge Question ID: 614

Question 9

For the following question, use technology to find the approximate Volume of the solid that results when the indicated areas are rotated about the y axis.

x=\tan y, x=0, y=1

Studyforge Question ID: 615

Question 10

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the line y=-1.

y=2\cos x,\ y=-1,\ x=-1,\ x=1

Studyforge Question ID: 616

Question 11

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the x axis.

y=\frac{1}{\sqrt{x^2+9}}, y=0,\ \ x=0\ to\ 3

Studyforge Question ID: 617

Question 12

For the following questions, find the Volume of the solid that results when the indicated areas are rotated about the y-axis.

x=1/\sqrt{4+y}, x=0, y\ from\ 0\ to\ 5\

Studyforge Question ID: 618

Question 13

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the y-axis.

x={\sec y,\ } x=0 ,\ y\ from-\frac{\pi }{4}\ to\frac{\pi }{4}

Studyforge Question ID: 619

Question 14

For the following question, find the Volume of the solid that results when the indicated areas are rotated about the y-axis.

x=y^3,\ \ y=x

Studyforge Question ID: 620

Question 15

For the following question, find the Volume of the solid that results when the indicated area is rotated about the y-axis.

y=e^{3x},\ x=0,\ x=1,\ y=0

Studyforge Question ID: 621

Question 16

Find the volume when a solid bounded by the curves y=3\sqrt{x-1}, y=0, and x=4 is revolved around the line y=-6.

Studyforge Question ID: 622

Question 17

Find the volume when a solid bounded by the curves y=3\sqrt{x-1} and y=0 and x=5 are revolved around the line x=-4.

Studyforge Question ID: 623

Question 18

Find the volume of a solid created by rotating the area bounded by x=2y-y^2, and x=y are revolved around the y-axis.

Studyforge Question ID: 624

Question 19

The region bounded between the curves x=2y^2 and y=2x is rotated about the line x=2. Using the disc/washer method (i.e. not cylindrical shells), determine the integral used to calculate the volume of the resulting solid. (Note: Set up the integral, but you do not have to evaluate.)

Studyforge Question ID: 625

Question 20

(Try this!) Find the volume of the sphere created by rotating the top half of the circle defined by the equation x^2+y^2=9 over the x-axis.

Studyforge Question ID: 626

Question 21

AP Prep:

What is the volume of the solid generated by rotating the area enclosed by f(x)=x^3, g(x)=2^{1-x}, the x-axis, and x=4, around the x-axis.

Studyforge Question ID: 9246

Question 22

AP Prep:

Find the volume of the solid generated by rotating the region bounded by the x-axis, y-axis, x=2,  f(x)=e^{x-1}, and g(x)=e^{1-x} around the x-axis.

Studyforge Question ID: 9247

Question 23

AP Prep:

Find the volume of a damaged cardboard box with a base we can model as the region under the parabola f(x)=\dfrac{(x-1)^2}{4}+1 bounded by the y-axis, the x-axis, and x=2, and cross sections perpendicular to the x-axis which are squares.

Studyforge Question ID: 9248

Question 24

AP Prep:

Find the volume of the solid generated by rotating the ellipse 4x^2+y^2=9 about the y-axis.

Studyforge Question ID: 9249

Question 25

AP Prep:

The base of a solid is the region enclosed by the parabola y = 3x^2, the line x = 2, and the x-axis. If the cross sections perpendicular to the x-axis are squares, find its volume.

Studyforge Question ID: 9604

Question 26

AP Prep:

The base of a solid is the region in the first quadrant enclosed by the graph y = 2e^{-x}, and the line x = 1. If the cross sections perpendicular to the x-axis are semi-circles, find its volume.

Studyforge Question ID: 9605

Question 27

AP Prep:

The base of a solid is the region enclosed by the graph of y = \sqrt{sin x}, the line y = 1, and the y-axis. If the cross sections perpendicular to the y-axis are equilateral triangles, find its volume.

Studyforge Question ID: 9606

Question 28

AP Prep:

The base of a solid is the region in the first quadrant bounded by the x-axis, y-axis and the line x + 4y = 8.If the cross sections perpendicular to the y-axis are rectangles whose width is half their length (where the length is in the x direction), find its volume.

Studyforge Question ID: 9607
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