# Integrating a Torus' Volume around the Y-axis Using Washer Method

• artikk
In summary, the conversation discusses finding the volume of a torus revolved around the y-axis using the washer method and the equation for a circle. The problem involves integrating (1-y^2)^1/2 dy from 1 to 0 multiplied by a constant, and one person suggests using a trigonometric substitution to solve the integral. The conversation concludes with a reminder to use the identity 1-sin^2(x)=cos^2(x) to simplify the integral.
artikk
I tried to find the volume of a torus revolved around the y-axis using the washer method. The equation for the circle is (x-2)^2 +y^2=1.
I need to integrate (1-y^2)^1/2 dy from 1 to 0 multiplied by a constant 16pi. The constant is not really important but I can't seem to integrate the integrand. I am aware that there's a formula for finding a vol. of a torus, as well.

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Unless you misstated the problem, it is trivial. The integral of 1 is y and the integral of y2 is y3/3. Just put them together.

Yes, but in this case, the (1-y^2) are under the square root, you can't separate the two terms.

$$\int_0^1 \sqrt (1-y^{2}) dy$$
a trig substitution would work here, like y=sinx, so dy=cos(x)dx, when y=0, x=0, when y=1,$$x=\frac{\pi}{2}$$

$$\int_0^\frac{\pi}{2} \sqrt(1-sin^{2}x )cosxdx$$ now remember

$$1-sin^{2}(x)=cos^{2}(x)$$ so

$$\int_0^\frac{\pi}{2} cos^{2}(x) dx=\int_0^\frac{\pi}{2} \frac{1+cos(2x)}{2} dx$$ and you are almost done!

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## 1. What is the washer method?

The washer method is a mathematical technique used to find the volume of a solid of revolution, such as a torus, when rotated around a certain axis. It involves breaking the solid into infinitesimally thin washers and calculating the volume of each one.

## 2. How do you integrate a torus' volume around the Y-axis?

To integrate a torus' volume around the Y-axis, you must use the washer method. This involves setting up an integral that represents the sum of all the infinitesimally thin washers that make up the torus. The integral will have limits of integration and a function representing the radius of each washer.

## 3. What is the formula for finding the volume of a washer?

The formula for finding the volume of a washer is V = π(Router2 - Rinner2)h, where Router is the outer radius of the washer, Rinner is the inner radius of the washer, and h is the height of the washer.

## 4. Can the washer method be used to find the volume of other shapes?

Yes, the washer method can be used to find the volume of other shapes that can be rotated around an axis, such as cones, cylinders, and spheres. However, the setup of the integral will be different for each shape.

## 5. Are there any limitations to using the washer method?

One limitation of the washer method is that it can only be used to find the volume of solids of revolution. It also requires a good understanding of integrals and how to set them up. Additionally, the washer method may be more difficult to use for more complex shapes with varying radii.

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