Washer Method Calculator

Washer method Calculator

Washer method calculator finds the volume of hollow cylinder-like shapes using integration. Get the step-wise solution of this procedure below the result.

How to use this calculator?

1. Enter the upper and lower bound limits.
2. Select the variable.
3. Input both of the functions.
4. Recheck the values in the “This will be calculated” area.
5. Click Calculate.

What is the washer method?

The washer method is a technique in calculus used to determine the volume of a solid of revolution when a region in the plane revolves around a line, resulting in a hollow or "washer"-shaped volume. 

The method involves integrating the difference between the squares of the outer and inner radii of these washers with respect to the axis of revolution.

In a more detailed explanation, if you have two functions y = f(x) and y = g(x) on an interval [a, b], where f(x)≥g(x)≥0 for all x in [a, b], and you revolve the region bounded by these functions around the x-axis, the volume V of the solid formed is:

V = π _a∫^b [f(x)]² −[g(x)]² dx

Here, 

• f(x) is the outer radius and 
• g(x) is the inner radius for each washer.

In practical terms, the washer method lets us find the volume of a "doughnut" shape (or any similar hollow shape) by stacking up many thin washers.

Why do we need the Washer method? 

There are formulas for the volumes of basic shapes like cylinders, spheres, cones, etc. But the washer method is particularly useful for finding volumes of more complicated shapes that can't be easily described by a single standard formula.

To be more specific, the formula for the volume of a basic cylinder is quite straightforward:

Volume = πr²h

Where:

• r is the radius of the cylinder.
• ℎ is the height of the cylinder.

For a sphere:

Volume = 4/3 πr³

However, for objects like a coffee cup, which isn't a perfect cylinder (due to its curved sides and potentially varying wall thickness), it doesn't fit neatly into the formula for cylinders. That's when methods from calculus, such as the washer method, become handy.

How to calculate volume using the washer method?

Here's how the washer method works:

Identifying the Radii: Imagine a thin strip in the region you're revolving. Once revolved around the line, this strip will form a "washer" (a disk with a hole in the middle). You'll want to identify the outer radius R(x) and the inner radius r(x) of this washer.

The volume of a Thin Washer: The volume dV of the thin washer formed by revolving this strip around the line is dV = π[R(x)² − r(x)²]dx where dx is the thickness of the strip.

Compute the Total Volume: Integrate the volume of the thin washer over the interval of interest to get the volume of the solid.

Example:

Find the volume of the solid generated by revolving the region bounded by the curves y = x² and y = 4 about the y-axis.

Visualization: Before solving, visualize the region. If you plot y = x², it's a parabola opening upwards. The line y = 4 is a horizontal line above the x-axis. 

The area between the curve and the line will look like an upside-down bowl when it revolves around the y-axis.

Solution: 

Step 1: Identify the bounds of integration

The region of interest is between the x-axis (where y = x² intersects it) and the line y = 4. For y = x² the intersection points are where x2 = 4 which are x = 2 and x = −2. These will be our bounds of integration: [−2,2].

Step 2: Set up the radii

Since we're revolving around the y-axis, the "radii" of our washers will be functions of y in terms of x.

1. Outer Radius R(y): This is the distance from the y-axis to the furthest curve from it. Here, for any value of y, that's the line on the right, and its distance from the y-axis is x. For the curve y = x², x = √y. Hence, R(y) = y.
2. Inner Radius r(y): This is the distance from the y-axis to the curve closest to it (on the left). But it's just the negative of the outer radius because the parabola is symmetric. So, r(y) = − y.

Step 3: Set up the integral for the volume.

The volume dV of a thin washer at height y with thickness dy is: 

dV = π[R(x)²− r(x)² ]dy

For this problem:

dV = π[√y²  - (-√y)² ]dy = π[y−y]dy = 0

Since the region is fully to the right of the y-axis, the inner radius is actually 0 (it's the y-axis itself).

So, the volume element is:

dV = π[√y²  - (0)² ]dy = π[y]dy = 0

Step 4: Integrate to find the volume

Now, integrate dV from y = 0 to y = 4:

V = ₀∫⁴πy dy = π₀∫⁴ y dy

V = π[ ½ y²]₀⁴

V = π[8−0] = 8π

So, the volume of the solid generated by revolving the given region about the y-axis is 8π cubic units.

Daily life Applications:

The washer method is a mathematical technique that has applications in various real-world scenarios where it's important to determine the volume of irregular shapes. Here are some daily life applications:

Manufacturing and Production:

• Containers: Companies that produce containers (like bottles, cups, or vases) with non-uniform shapes might use principles similar to the washer method to determine the volume capacity of these containers.
• Machinery Parts: For custom-made parts that need a particular volume (for example, in automotive or aerospace industries), the washer method might be applied.

Civil Engineering and Architecture:

• Structural Design: Architects and engineers can use the washer method to determine the volume of materials needed for uniquely shaped structures, like domes or innovative building designs.

Medical Imaging:

• Organ Volume Estimation: In medical imaging, particularly when viewing cross-sectional images like CT scans or MRIs, the volume of organs or tumors that don't have simple geometric shapes can be estimated using principles akin to the washer method.