1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
amid [387]
3 years ago
14

A torus is formed when a circle of radius 3 centered at (5 comma 0 )is revolved about the​ y-axis. a. Use the shell method to wr

ite an integral for the volume of the torus. b. Use the washer method to write an integral for the volume of the torus. c. Find the volume of the torus by evaluating one of the two integrals obtained in parts​ (a) and​ (b). (Hint: Both integrals can be evaluated without using the Fundamental Theorem of​ Calculus.)
Physics
1 answer:
arlik [135]3 years ago
8 0

Answer:

a) V=4\pi\int\limits^8_2 {x\sqrt{9-(x-5)^{2}}} \, dx

b) V=20\pi\int\limits^3_{-3} {\sqrt{9-y^{2}}} \, dy

c) V=90\pi ^{2}

Explanation

In order to solve these problems, we must start by sketching a drawing of what the graph of the problem looks like, this will help us analyze the drawing better and take have a better understanding of the problem (see attached pictures).

a)

On part A we must build an integral for the volume of the torus by using the shell method. The shell method formula looks like this:

V=\int\limits^a_b {2\pi r y } \, dr

Where r is the radius of the shell, y is the height of the shell and dr is the width of the wall of the shell.

So in this case, r=x so dr=dx.

y is given by the equation of the circle of radius 3 centered at (5,0) which is:

(x-5)^{2}+y^{2}=9

when solving for y we get that:

y=\sqrt{9-(x-5)^{2}}

we can now plug all these values into the shell method formula, so we get:

V=\int\limits^8_2 {2\pi x \sqrt{9-(x-5)^{2}} } \, dx

now there is a twist to this problem since that will be the formula for half a torus.Luckily for us the circle is symmetric about the x-axis, so we can just multiply this integral by 2 to get the whole volume of the torus, so the whole integral is:

V=\int\limits^8_2 {4\pi x \sqrt{9-(x-5)^{2}} } \, dx

we can take the constants out of the integral sign so we get the final answer to be:

V=4\pi\int\limits^8_2 {x\sqrt{9-(x-5)^{2}}} \, dx

b)

Now we need to build an integral equation of the torus by using the washer method. In this case the formula for the washer method looks like this:

V=\int\limits^b_a{\pi(R^{2}-r^{2})} \, dy

where R is the outer radius of the washer and r is the inner radius of the washer and dy is the width of the washer.

In this case both R and r are given by the x-equation of the circle. We start with the equation of the circle:

(x-5)^{2}+y^{2}=9

when solving for x we get that:

x=\sqrt{9-y^{2}}+5

the same thing happens here, the square root can either give you a positive or a negative value, so that will determine the difference between R and r, so we get that:

R=\sqrt{9-y^{2}}+5

and

r=-\sqrt{9-y^{2}}+5

we can now plug these into the volume formula:

V=\pi \int\limits^3_{-3}{(5+\sqrt{9-y^{2}})^{2}-(5-\sqrt{9-y^{2}})^{2}} \, dy

This can be simplified by expanding the perfect squares and when eliminating like terms we end up with:

V=20\pi\int\limits^3_{-3} {\sqrt{9-y^{2}}} \, dy

c) We are going to solve the integral we got by using the washer method for it to be easier for us to solve, so let's take the integral:

V=20\pi\int\limits^3_{-3} {\sqrt{9-y^{2}}} \, dy

This integral can be solved by using trigonometric substitution so first we set:

y=3 sin \theta

which means that:

dy=3 cos \theta d\theta

from this, we also know that:

\theta=sin^{-1}(\frac{y}{3})

so we can set the new limits of integration to be:

\theta_{1}=sin^{-1}(\frac{-3}{3})

\theta_{1}=-\frac{\pi}{2}

and

\theta_{2}=sin^{-1}(\frac{3}{3})

\theta_{2}=\frac{\pi}{2}

so we can rewrite our integral:

V=20\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {\sqrt{9-(3 sin \theta)^{2}}} \, 3 cos \theta d\theta

which simplifies to:

V=60\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {(\sqrt{9-(3 sin \theta)^{2}}} \, cos \theta d\theta

we can further simplify this integral like this:

V=60\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {(\sqrt{9-9 sin^{2} \theta}}} \, cos \theta d\theta

V=60\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {3(\sqrt{1- sin^{2} \theta})}} \, cos \theta d\theta

V=180\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {(\sqrt{1- sin^{2} \theta})}} \, cos \theta d\theta

V=180\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {(\sqrt{cos^{2} \theta})}} \, cos \theta d\theta

V=180\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {(cos \theta})} \, cos \theta d\theta

V=180\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {cos^{2} \theta}} \, d\theta

We can use trigonometric identities to simplify this so we get:

V=180\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {\frac{1+cos 2\theta}{2}}} \, d\theta

V=90\pi\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}} {1+cos 2\theta}}} \, d\theta

we can solve this by using u-substitution so we get:

u=2\theta

du=2d\theta

and:

u_{1}=2(-\frac{\pi}{2})=-\pi

u_{2}=2(\frac{\pi}{2})=\pi

so when substituting we get that:

V=45\pi\int\limits^{\pi}_{-\pi} {1+cos u}} \, du

when integrating we get that:

V=45\pi(u+sin u)\limit^{\pi}_{-\pi}

when evaluating we get that:

V=45\pi[(\pi+0)-(-\pi+0)]

which yields:

V=90\pi ^{2}

You might be interested in
After watching a video about submarines, Jamil wants to learn more about the ocean. which question could be answered through sci
SOVA2 [1]
<span>Submarines are equipped with water tanks called ballast tanks that fill up to submerge the vessel. Emptying the tanks and filling them with air causes the submarine to surface.</span>
5 0
3 years ago
During its lifespan, what characteristics of the sun will change
4vir4ik [10]
During its lifepsan, the sun's core would keep contracting and heating up.

The temperature will keep increasing to the point where the temperature outside the core will get to hydrogen fusion temperatures.
The sun will grow in surface and eventually became the Red Giant
6 0
3 years ago
Read 2 more answers
You shake a bottle of soda and take off the cap. If the soda shoots out of the
Harman [31]

Answer:

A. 16.9 m

Explanation:

I think this is the answer i am not sure

but hope it helps

3 0
3 years ago
Read 2 more answers
Why do the waves have different speeds in different layers of Earth's surface?
Genrish500 [490]

Answer:

Material's density

Explanation:

Seismic waves travel at different rates of speed based on a material's density. Hopefully, you understand that the Earth has three main layers: the crust, mantle, and core. Earthquake waves move faster through solids.

6 0
2 years ago
Why was Dalton's atomic theory difficult for other scientists to accept? please help?!!!!!
Debora [2.8K]
It is cellmicktomic atoms
7 0
3 years ago
Other questions:
  • What term is used to describe the hot ash, gases, and materials being erupted at speeds of up to 400 mph?
    11·1 answer
  • Jasper and Gemma are going to play on a teeter totter. Gemma gets on first. When Jasper gets on, Gemma moves into the air, but s
    5·2 answers
  • A rock is thrown upward from level ground in such a way that the maximumheight of its flight is equal to its horizontal rangeR.
    6·1 answer
  • If one gram of matter could be completely converted into energy, the yield would be
    7·1 answer
  • If an engine has a compression ratio of 7:1,
    7·2 answers
  • A stone dropped from the top of a 80m high building strikes the ground at 40 m/s after falling for 4 seconds. The stone's potent
    6·1 answer
  • Pls help I will give brainlist and don’t give me a link I can’t open them
    8·1 answer
  • True or False: The medium moves up and down in a transverse wave.
    5·1 answer
  • Define what is false​
    10·1 answer
  • 16) A car's position in relation to time is plotted on the graph. What can be said about the car during segment B?
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!