How to find the Volume of Solid of Revolution

Learn The Different Ways To Find The Volume Of Solid Of Revolution By Various Methods In Integration.

Alan Walker-

Published on 2023-04-13

Introduction

We can find the volume obtained by rotation by using the concept of indefinite integral. The volume of solid revolution is obtained from an indefinite integral if it revolve around a plane region. But for this to happen, the line must not pass through that plane. So there are three methods to do this type of revolution around a vertical or a horizontal line. These methods are:

1. Disk Method
2. Washer Method
3. Shell Method

In this article we will explain different methods used for finding volume of solid revolutions and the difference between three of them. This will make you understand when to use each these methods.

Related: Learn about different types of integration methods and how to understand integration of partial fraction quickly.

How to determine which Method to Use for Volume?

If you want to determine which method to use for volume of solid revolutions, it depends on the type of rotation evolving in it. The below flowchart help you to analyze which method is to use:

Now this raises a question, Is the axis of revolution parallel or perpendicular?

Best way to deal with this is that if it's parallel, we can use shell method to find the volume.

But in case if it is perpendicular, it raises another question. Does the axis of revolution is on the boundary of the plane region?

If axis of rotation is at boundary of the plane we should use the disk method to find the volume of solid revolution and if axis of rotation is not at boundary of the plane then use should use washer method to find the volume of solid of revolution.

Now one more question arise that which method is best fit in our problems. So before we go through qurstion problems we may concern when to use washer or shell method for finding volume.

These questions makes it easy for someone to understand which method is fit to use. Now, let's find the difference between these methods.

Related: What are the benefits of using integral calculator?

Disk Method

The disk method is the method for calculating the volume of solid-state material which revolve around the axis. This may be along the x-axis or y-axis depending on the requirement.

Disc method is used where the axis of rotation is attached with the boundary of the plane region. It becomes easy to calculate the volume of a solid around a line or its axis of rotation by using disc method.

In the Disk method Integration, if we create a three-dimensional shape of a solid region obtained by the revolution of our function. And divide it into an infinite number of discs of different radius having infinitesimal thicknesses, we will be able to find the volume of each disc with a different radius.

You can also use integration by parts calculator with steps on home page of this integral website.

Rotation along X-axis

If the function to be revolved is along the x-axis, then integral represents the volume of the solid of revolution:

$V \;=\; \int_a^b (π R^2) (w) {2}lt;/p>$ \text{Or, } V \;=\; \int_a^b π f(x)^2 \; (Δx) {2}lt;/p>

$V \;=\; \int_a^b π f(x)^2 \; dx{2}lt;/p> Rotation along Y-axis If the function to be revolved is along the y-axis, then integral represents the volume of the solid of revolution:$ V \;=\; \int_a^b (π R^2) (w) {2}lt;/p>

$\text{Or, } V \;=\; \int_a^b π f(y)^2 \; (Δy) {2}lt;/p>$ V \;=\; \int_a^b π f(y)^2 \; dy{2}lt;/p>

Here f(x) and f(y) show the radii of the disks we have created in solid three dimensional shape or the distance between the axis of revolution and a point on the curve.

Now we will understand this concept in a practical way with detailed example

Example

Let's find the volume of the solid obtained by rotating the region under the graph y = x 2 about the x-axis over the interval of [−2,3].

Solution:

Since the x-axis is the boundary point of the region. So disc method is used here and the graph of such function for obtaining solid region is given as:

The volume of a solid revolution by disk method is calculated as:

$V \;=\; \int_{-2}^3 π (x^2)^2 dx {2}lt;/p>$ V \;=\; π \int_{-2}^3 x^4 dx {2}lt;/p>

$V \;=\; π \left[ \frac{1}{5}x^5 \right]_{-2}^3 {2}lt;/p>$ V \;=\; π \left[ \frac{243}{5} \;-\; \left( \frac{-32}{5} \right) \right] {2}lt;/p>

Rotation along y-axis

Similarly, the volume(V) of the solid generated by rotating the curve between f(x) and g(x) on an interval of [a,b] about the y-axis is given as:

$V \;=\; \int_a^b π \left( [f(y)]^2 \;-\; [g(y)]^2 \right) dy{2}lt;/p> Here, f(x),g(x),f(y) and g(y) represent the outer radii and inner radii of the washer volume. Now we will understand in a practical way with a detailed example Example Let's find the volume of the solid generated by rotating the curve between y = x2+2 and y = x+4 about the x-axis. Solution: Since, y = x2+2 and y = x+4, So x2+2 = x+4 x2-x-2 = 0 (x+1)(x-2) = 0 x = -1 and x = 2 From that, the graph of the above functions will intersect at (-1,3) and (2,6), It will be given as: The volume of a solid revolution by washer method is calculated as:$ V \;=\; \int_{-1}^2 π \left[ (x+4)^2 - (x^2 + 2)^2 \right] dx {2}lt;/p>

$V \;=\; \int_{-1}^2 π \left[ (x^2 + 8x + 16) - (x^4 + 4x^2 + 4) \right] dx {2}lt;/p>$ V \;=\; π \int_{-1}^2 (-x^4 - 3x^2 + 8x + 12) dx {2}lt;/p>

$V \;=\; π \left[ -\frac{1}{5} x^5 - x^3 + 4x^2 + 12x) \right]_{-1}^{2} {2}lt;/p>$ V \;=\; π \left[ \frac{128}{5} - \left( - \frac{34}{5} \right) \right] {2}lt;/p>

Rotation along y-axis

The Volume(V) of the solid is obtained by rotating the region x = f(y) when rotated about the y-axis on the interval of [a,b], then the volume is:

$V \;=\; \int_a^b 2πy \; f(y) \; dy {2}lt;/p> Here the x and y under the integral (integrand) are the radii of the shell method, while on the other hand f(x) and f(y) represent the height of the shell method. Let’s move on to an example for a better understanding of the Shell method to find the volume of solid. This website help you solve indefinite integral online and definite integral online by using online calculators. Example Find the volume of solid obtained by rotating the region bounded by y = x2 and x-axis[1,3] when rotated about the y-axis. Solution: Since axis of rotation is vertical in shell method, so it will be expressed in terms of x i.e radius of shell is “x” and height of the shell is “f(x) = x^2” as given in a figure: The volume of a solid revolution by cylindrical shell method is calculated as:$ V \;=\; \int_1^3 2πx \; x^2 dx {2}lt;/p>

$V \;=\; 2π \int_1^3 x^3 dx {2}lt;/p>$ V \;=\; 2π \left[ \frac{1}{4}x^4 \right] {2}lt;/p>

$V \;=\; \frac{π}{2} (81-1) {2}lt;/p>$ V \;=\; 40π {2}lt;/p>

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