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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