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The calculator determines the accurate volume of a revolution. Enter the values for f(x) , g(x), Upper & Lower limits to get complete steps involved.

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Calculate the volume of a solid rotation with washer method calculator. The tool uses method of definite integration and determines the revolution of the function around the axes that is the entered variable.

Get step wise solution to integrate the volume and understand the problem better.

**"Washer method helps you calculate the volume of a function rotation about a given axes by integrating it"**

The single washer volume formula is:

\(V = π (r_2^2 – r_1^2) h = π (f (x)^2 – g (x)^2) dx\)

The exact volume formula arises from taking a limit as the number of slices becomes infinite.

Formula for washer method

\(V = π ∫_a^b [f (x)^2 – g (x)^2] dx\)

Find the volume of the solid when

Top curve = y = x and

Bottom one = y = \(x^2\)

This is definitely a complete revolution. We will set up a formula where

- f(x) = x and g(x) = \(x^2\)

But what should we use for the point a and b?

Well, this area is limited by the two curves between its common intersection.

**Steps:**

Set **f(x)** to **g(x)** and find to locate the points of intersections.

Substitute the a = 0 and b = 1 in the washer method equation.

- \(V = π ∫_a^b [f (x)^2 – g (x)^2] dx\)
- \(V = π ∫_0^1 [ (x)^2 – (x^2)^2] dx\)
- \(V = π ∫_0^1 [ (x)^2 – (x)^4] dx\)
- \(V = π ∫_0^1 [(x)^2 – (x)^4] dx\)
- \(V = π [(x)^3 / 3 – (x)^5 / 5]^1_0\)
- \(V = 2 π / 15\)

- First, Enter the value for the functions f (x) and g (x)
- Now, substitute the upper and lower limit.
- Hit the calculate button for results.

- The calculator provides the definite and indefinite integral of given functions
- You get step-by-step calculations with different methods

If it is parallel to the slices, then every slice will trace out a cylindrical shell as it revolves around the axis.

On the other hand, if it's perpendicular to the slices, every slice will trace out a disk or washer as it revolves around the axis.

Wikipedia: Disc integration, Washer method, the axis of revolution.

Magoosh: Solids of Revolution, The Disk and Washer Methods: Formulas, Different Axes.

Dummies: Volume of a Shape Using the Washer Method, Outer and Inner Radius.

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