Gunsaz

2022-09-07

Volume of Solid of Revolution about an equation

I learned about the disk and shell method for finding a volume of a solid of revolution. For my question I don't think I can use either method directly. Here is my question:

Find the volume of the region bounded by $y=x$ and $y={x}^{2}$, but rotated about the equation of $y=x$.

Here the axis of revolution is not a vertical or horizontal line, but rather the equation $y=x$. One idea I had was to convert this diagonal line into a horizontal or vertical one, but I can't see to do this. Maybe polar coordinates might be useful, but I'm not sure.

I learned about the disk and shell method for finding a volume of a solid of revolution. For my question I don't think I can use either method directly. Here is my question:

Find the volume of the region bounded by $y=x$ and $y={x}^{2}$, but rotated about the equation of $y=x$.

Here the axis of revolution is not a vertical or horizontal line, but rather the equation $y=x$. One idea I had was to convert this diagonal line into a horizontal or vertical one, but I can't see to do this. Maybe polar coordinates might be useful, but I'm not sure.

emarisidie6

Beginner2022-09-08Added 7 answers

Step 1

Another approach using a ''modified'' disk method.

The distance of a point $(x,y)=(x,{x}^{2})$ from the line $y=x$ is:

$r=\frac{|x-{x}^{2}|}{\sqrt{2}}$

so the area of a circle orthogonal to the axis of rotation is

$A=\pi {r}^{2}=\pi \frac{(x-{x}^{2}{)}^{2}}{2}$

Step 2

And the volume of a disk of length $\delta x=\sqrt{2}dx$ along the axis of rotation is

$dV=\frac{\pi}{2}{\int}_{0}^{1}(x-{x}^{2}{)}^{2}\delta x=\frac{\pi}{\sqrt{2}}{\int}_{0}^{1}(x-{x}^{2}{)}^{2}dx$

And the volume of the solid of revolution is the integral:

$V=\frac{\pi}{\sqrt{2}}{\int}_{0}^{1}(x-{x}^{2}{)}^{2}dx$

Another approach using a ''modified'' disk method.

The distance of a point $(x,y)=(x,{x}^{2})$ from the line $y=x$ is:

$r=\frac{|x-{x}^{2}|}{\sqrt{2}}$

so the area of a circle orthogonal to the axis of rotation is

$A=\pi {r}^{2}=\pi \frac{(x-{x}^{2}{)}^{2}}{2}$

Step 2

And the volume of a disk of length $\delta x=\sqrt{2}dx$ along the axis of rotation is

$dV=\frac{\pi}{2}{\int}_{0}^{1}(x-{x}^{2}{)}^{2}\delta x=\frac{\pi}{\sqrt{2}}{\int}_{0}^{1}(x-{x}^{2}{)}^{2}dx$

And the volume of the solid of revolution is the integral:

$V=\frac{\pi}{\sqrt{2}}{\int}_{0}^{1}(x-{x}^{2}{)}^{2}dx$

rialsv

Beginner2022-09-09Added 3 answers

Step 1

Start by writing the curve in parametric form, so $x=t,y={t}^{2}$. Now use a matrix to rotate the curve by 45 degrees clockwise so that the new curve is also given parametrically.

$(}\genfrac{}{}{0ex}{}{x}{y}{\textstyle )}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ -1& 1\end{array}\right){\textstyle (}\genfrac{}{}{0ex}{}{t}{{t}^{2}}{\textstyle )$

Step 2

Now consider the volume as

$\pi {\int}_{t=0}^{1}{y}^{2}\frac{dx}{dt}dt$

Start by writing the curve in parametric form, so $x=t,y={t}^{2}$. Now use a matrix to rotate the curve by 45 degrees clockwise so that the new curve is also given parametrically.

$(}\genfrac{}{}{0ex}{}{x}{y}{\textstyle )}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ -1& 1\end{array}\right){\textstyle (}\genfrac{}{}{0ex}{}{t}{{t}^{2}}{\textstyle )$

Step 2

Now consider the volume as

$\pi {\int}_{t=0}^{1}{y}^{2}\frac{dx}{dt}dt$

The distance between the centers of two circles C1 and C2 is equal to 10 cm. The circles have equal radii of 10 cm.

A part of circumference of a circle is called

A. Radius

B. Segment

C. Arc

D. SectorThe perimeter of a basketball court is 108 meters and the length is 6 meters longer than twice the width. What are the length and width?

What are the coordinates of the center and the length of the radius of the circle represented by the equation ${x}^{2}+{y}^{2}-4x+8y+11=0$?

Which of the following pairs of angles are supplementary?

128,62

113,47

154,36

108,72What is the surface area to volume ratio of a sphere?

An angle which measures 89 degrees is a/an _____.

right angle

acute angle

obtuse angle

straight angleHerman drew a 4 sided figure which had only one pair of parallel sides. What could this figure be?

Trapezium

Parallelogram

Square

RectangleWhich quadrilateral has: All sides equal, and opposite angles equal?

Trapezium

Rhombus

Kite

RectangleKaren says every equilateral triangle is acute. Is this true?

Find the number of lines of symmetry of a circle.

A. 0

B. 4

C. 2

D. InfiniteThe endpoints of a diameter of a circle are located at (5,9) and (11, 17). What is the equation of the circle?

What is the number of lines of symmetry in a scalene triangle?

A. 0

B. 1

C. 2

D. 3How many diagonals does a rectangle has?

A quadrilateral whose diagonals are unequal, perpendicular and bisect each other is called a.

A. rhombus

B. trapezium

C. parallelogram