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Math Calculators ▶ Washer Method Calculator

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An online washer method calculator allows you to determine the volume of a solid of revolution to cover the solids with a hole. This washer calculator finds the definite and indefinite integration of the functions f(x) and g(x). Here you can learn how to do washer method with the help of a washer formula.

This process is a logical extension of the disk method for determining the volumes of the solids of revolution. Use the Washer Method to set up the definite and indefinite integral that provides the volume of the revolution.

A washer is the same as a disk but with a center, the hole cut out. The formula of volume of a washer requires both an outer radius r^1 and an inner radius r^2.

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

**Example: **

Find the volume of the solid, when the bounding curves for creating the region are outlined in red. The top curve is y = x and bottom one is y = x^2

**Solution: **

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, with some area problems, you may need to find limitations. Obviously, this area is limited by the two curves between its common intersection. 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.

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

$$ V = π ∫_0^1 [ (x)^2 – (x^2)^2] dx $$

$$ V = π ∫_0^1 [ (x)^2 – (x)^4] dx $$

$$ V = π ∫_0^1 [(x)^2 – (x)^4] dx $$

$$ V = π [(x)^3 / 3 – (x)^5 / 5]^1_0 $$

$$ V = 2 π / 15 $$

An online washer calculation finds the definite and indefinite integral of the volume of revolution

- 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 washer method calculator provides the definite and indefinite integral of given functions.
- The calculator does step-by-step calculations with different methods.

If it’s parallel to the slices, then every slice will trace out a cylindrical shell as it revolves around the axis. On the other hand, it’s perpendicular to the slices, every slice will trace out a disk or washer as it revolves around the axis.

The Washer method is just like the Disk method when the inner disk is subtracted from the outer disk.

Use this washer method calculator for determining where two different curves intersect each other. The calculator takes the definite and indefinite integral of functions with different methods. Undoubtedly, the calculations of the washer method are harder to do, to make it convenient for you, our free online calculator does all washer-related calculations for you in a fraction of a second.

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

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

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