Of the shape shown below.<br> units?<br> Show Calculator<br> What is the answer

A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R (> r) from th

jeyben [28]

Consider a circle with radius $r$ centered at some point $(R+r,0)$ on the $x$ -axis. This circle has equation

$(x-(R+r))^2+y^2=r^2$

Revolve the region bounded by this circle across the $y$ -axis to get a torus. Using the shell method, the volume of the resulting torus is

$\displaystyle2\pi\int_R^{R+2r}2xy\,\mathrm dx$

where $2y=\sqrt{r^2-(x-(R+r))^2}-(-\sqrt{r^2-(x-(R+r))^2})=2\sqrt{r^2-(x-(R+r))^2}$ .

So the volume is

$\displaystyle4\pi\int_R^{R+2r}x\sqrt{r^2-(x-(R+r))^2}\,\mathrm dx$

Substitute

$x-(R+r)=r\sin t\implies\mathrm dx=r\cos t\,\mathrm dt$

and the integral becomes

$\displaystyle4\pi r^2\int_{-\pi/2}^{\pi/2}(R+r+r\sin t)\cos^2t\,\mathrm dt$

Notice that $\sin t\cos^2t$ is an odd function, so the integral over $\left[-\frac\pi2,\frac\pi2\right]$ is 0. This leaves us with

$\displaystyle4\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}\cos^2t\,\mathrm dt$

Write

$\cos^2t=\dfrac{1+\cos(2t)}2$

so the volume is

$\displaystyle2\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}(1+\cos(2t))\,\mathrm dt=\boxed{2\pi^2r^2(R+r)}$

6 0

3 years ago

This is the question.

umka21 [38]

Answer:

Step-by-step explanation:

as you can see they want you to find the total out of the gallons so the first thing you want to do is get 1,688 and 1,297 and multiply both of them the total is 2189336 so 1,688 x 1,297=289336

4 0

3 years ago

An imaginary number has which of the following as a definition?

lyudmila [28]

Her is the definition of an imaginary number An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.
(Got that from google)

3 0

4 years ago

The sum of two numbers is 30 and their difference is 12

Elina [12.6K]

The correct answer is 21 and 9. 21+9=30 and 21-9=12. Hope this helps.

6 0

3 years ago

A map shows the distance between the corner of Cactus Road and 1st Street and the corner of 1st Street and Merle Road as 3 inche

oksano4ka [1.4K]

C: 11.1 mi Just do 3.7*3 to get the answer which is from the scale 1:3.7.

6 0

3 years ago

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Of the shape shown below. units? Show Calculator What is the answer