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