Applications of Integration – Volumes: Solids of Revolution – Disk Method – Method of Cylinders

In geometry, we learned the formulas for the volume of several figures, such as rectangular prisms, pyramids, and cylinders (solid and hollow). For this section, we will focus on the volume of a solid cylinder, how the formula was derived, and how we can use the same procedure to derive the volume of any shape given a function.

Suppose we have a function, which is a line on the coordinate plane y = r (the radius) which extends from x = 0 to x = h. The length of the line on the interval [0,h] is the height: h – 0 = h.
Afterwards, imagine the line y = r being rotated about the x axis. When the line makes a complete revolution, a cylinder forms:

You can imagine that a cylinder is a collection of circles which, if stacked on top of each other, will form a cylinder with height h. That means we have to find the area of each circle (Acircle= πr2), and add them up.

We can find the sum of all the circles by taking the integral of the area function on the interval x = 0 to x = h, integrating:

Since r2 is a constant, it will be factored from the integrand:

Integrate the constant of 1:

In this way, we have derived the volume of a cylinder: r2h
We can apply the derivation of a solid cylinder to find the volume of a solid of revolution.

Formula: Solid of Revolution – Disk Method

Suppose you have a curve y = F(x) (which will serve as the ‘radius’ of the solid of revolution) that extends from x =a to x = b:

After rotating it along the x-axis, a cylinder-like figure will form. Notice how the cross section of the solid looks like a circle:

If the radius is r = F (x), the volume that will be produced by rotating the curve around x-axis is,

Of course, if we are rotating a function x = F(y) about the y axis, the volume of the revolved region will be:

Guidelines for finding a Solid of Revolution

1. Draw the curves on the coordinate plane. Determine axis of rotation (with respect to x or y).
2. Determine the limits of integration.
3. Apply the formula

Example 1: Find the volume of revolution of the enclosed region y = -x2+4 and y = 0.

Solution:
Draw the curves on the coordinate plane. Determine axis of rotation (with respect to x or y).

y = -x2+4 and y = 0, as well as the shaded region bounded by both curves, are drawn below:

By looking at the diagram, the axis of rotation is y = 0, or the x axis. Therefore, we rotate with respect to the x axis.

Determine the limits of integration.

In this case, we need to find the points of intersection:

0 = -x2+4
x2= 4
x = +/- 2

The limits of integration are a = -2 and b = 2.

Apply the formula.

Our radius R(x) is y = -x2+4, which can be determined from drawing an arrow from the axis of rotation to the topmost curve.

Therefore, using the formula,

The volume of the solid of revolution is shown below:

Example 2: Find the volume of revolution of the enclosed region y = \sqrt{x}+2 , y = 2, and x = 4.

Solution:
Draw the curves on the coordinate plane. Determine axis of rotation (with respect to x or y).

The function y = \sqrt{x}+2, y = 2, and x = 4, as well as the region bound by both curves are drawn below:

From the diagram, the axis of rotation will be y = 2 (which is parallel to the x axis). Therefore, we will be rotating with respect to the x axis.

Determine the limits of integration.

The limits of integration will be along the axis of rotation.
We see that the volume will be formed between x = 0 and x = 4. Therefore, the limits of integration are a = 0 and b = 4.

Apply the formula.

Our radius R is the difference between the top curve y = \sqrt{x}+2 and axis of rotation y = 2:

Using the formula,

The volume of the solid of revolution is 8⌅ cubic units. The solid of the revolution is shown below:

Example 3: Find the volume of revolution of the enclosed region x = \sqrt{y+5}, x = 0, y = -1, and y = 5 about the y axis.
Solution:

Draw the curves on the coordinate plane. Determine axis of rotation (with respect to x or y).

The function x = \sqrt{y+5}, x = 0, y = -1, and y = 5, as well as the region bound by the curves, are drawn below:

The area bound by the curves is shaded below:

As indicated in the problem, we will rotate the solid about the Y axis (x = 0). Therefore, we are revolving with respect to y.

Determine the limits of integration.

The limits of integration are along the axis of rotation.
We see that the volume will be formed between y = -1 and y = 5. Therefore, the limits of integration are c = -1 and d = 5.

Apply the formula.

The radius R(y) is the function x = \sqrt{y+5}, which can be determined by drawing an arrow from the axis of rotation to the rightmost curve:

Therefore,

The volume of revolution is shown below. In this diagram, the y axis is the horizontal axis, and the x axis is the vertical axis.

Example 4: Find the volume of revolution of the enclosed region x = y2+1, x = -1, y = -2, and y = 1 about x = -1
Solution:
Draw the curves on the coordinate plane. Determine axis of rotation (with respect to x or y).

The function x = y2+1, x = -1, y = -2, and y = 1, as well as the region bound by these curves, are shaded below:

As indicated in the problem, the axis of rotation is x = -1 (which is parallel to the y axis), therefore, we will rotate with respect to the y axis.

Determine Limits of Integration

The limits of integration will be along the y axis. By looking at the graph, the solid is bounded between y = -2 and y = 1. Therefore, c = -2 and d = -1.

Apply the formula

The radius of the function R(y) is x = y2+2, can be determined by drawing an arrow from the axis of rotation to the rightmost curve:

Therefore,

The volume of revolution is shown below. In the diagram, the y axis is the horizontal axis, and the x axis is the vertical axis:

Example 5: Find the volume of revolution of the enclosed region bounded by x = 10 and x = y2-1 revolved around x = 10.
Solution:

Draw the curves on the coordinate plane. Determine axis of rotation (with respect to x or y).

The curves x = 10 and x =y2-1 , as well as the region bounded by the curves, are drawn below:

The region enclosed by the curves is shaded below:

As indicated in the problem, we are revolving around the vertical axis x = 10 (which is parallel to the y axis), so we are revolving with respect to y.

Determine limits of integration

Since we are revolving around a vertical axis (x = 10), the limits of integration are along the axis of rotation.
Find the points of intersection (set the equation of each curve equal to each other) to find the lower and upper limit:

10 = y2-1
11 = y2
y = +/- 3.317

Therefore, d = -3.317 and d = 3.317.

Apply the formula

The radius of the solid of revolution is R(y) = 10 – (y2-1) = 11-y2, which can be determined by drawing an arrow from the axis of rotation to the leftmost curve:

Therefore,

The solid of revolution is shown below. In the diagram, the y axis is the horizontal axis and the x axis is the vertical axis:

Example 6: Find the volume of revolution of the enclosed region bounded by x = 0, y = 1, and y = 2x1/3-2revolving around y = 1
Solution:

Draw the curves on the coordinate plane. Determine axis of rotation (with respect to x or y).

The curves x = 0, y = 1, and y = 2x1/3-2, as well as the region bounded by the curves are drawn on the coordinate plane below:

As indicated in the problem, the axis of rotation is y = 1 (which is parallel to the x axis). Therefore, we will rotate with respect to the x axis.

Determine the limits of integration

The limits of integration will be along the axis of rotation (y = 1). From the diagram, on the left side, the lower limit is x = 0. On the right side, the point of intersection of y = 1 and y = 2x1/3-2is the upper limit:

1 = 2x1/3-2
3 = 2x1/3
3/2= x1/3
(3/2)3/1=(x1/3)3/1
(3/2)3/1=x
3.375 = x

Therefore, a = 0 and b = 3.375

Apply the formula

The radius of the solid of revolution is R(x) = 1 – (2x1/3-2) = 3 – 2x1/3, which can determined by drawing an arrow from the axis of rotation to the bottommost curve:

Therefore,

The volume of revolution is shown below:

Summary of Section

A solid of revolution is a volume obtained by rotating a planar arc around the axis of revolution that lies on the same plane.

The disk method is used to calculate the volume of a solid of revolution when integrating along an axis parallel to the axis of revolution.
If the function being revolved is a function of x and is revolved around the x axis, the solid of revolution is:

If the function being revolved is a function of y and is revolved around the y axis, the solid of revolution is:

References:

https://www.math24.net/volume-solid-of-revolution-disks-washers/#example5
http://scidiv.bellevuecollege.edu/dh/Calculus_all/CC_5_1_Volumes.pdf
http://www.opentextbookstore.com/appcalc/Chapter3-6.pdf

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