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4. Volume of Solid of Revolution

by M. Bourne

Many solid objects, especially those made on a lathe, have a circular cross-section and curved sides. In this section, we see how to find the volume of such objects using integration.

Example 1

Consider the area bounded by the straight line y = 3x the x-axis, and x = 1:

 

When the shaded area is rotated 360° about the x-axis, we observe that a volume is generated in this LiveMath animation:

LIVEMath

 

The resulting solid is a cone:

 

Disk Method for Finding Volumes

To find this volume, we could take slices (the yellow disk shown above), each dx wide and radius y:

The volume of a cylinder is given by:

V = πr²h

Because radius = r = y and each disk is dx high, we notice that the volume of each slice is:

V = πy² dx

Adding the volumes of the disks (with infinitely small dx), we obtain the formula:

 which means 

where:

y = f(x) is the equation of the curve whose area is being rotated

a and b are the limits of the area being rotated

dx shows that the area is being rotated about the x-axis

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NOTE: We use the disk method only (not Shell Method) in this section.

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Applying the formula to the earlier example, we have:

 

CHECK: Does the method work? We can find the volume of the cone using

(Checks OK.)

 

Example 2

Find the volume if the area bounded by the curve y = x³ + 1, the x-axis and the limits of x = 0 and x = 3 is rotated around the x-axis.

Here's what it looks like using LiveMath:

LIVEMath

Now for the normal answer:

Answer

Volume by Rotating the Area Enclosed Between 2 Curves

If we have 2 curves y₂ and y₁ that enclose some area and we rotate that area around the x-axis, then the volume of the solid formed is given by:

In the following general graph, y₂ is in blue and y₁ is black. The lower and upper limits for the region to be rotated are indicated in dark red: x = a to x = b.

Example 3

A cup is made by rotating the area between y = 2x² and y = x + 1 with x ≥ 0 around the x-axis. Find the volume of the material needed to make the cup. Units are cm.

Answer

Rotation around the y-axis

When the shaded area is rotated 360° about the y-axis, the volume that is generated can be found by:

 which means 

where:

x =f(y) is the equation of the curve expressed in terms of y

c and d are the upper and lower y limits of the area being rotated

dy shows that the area is being rotated about the y-axis

 

Example 4

Find the volume of the solid of revolution generated by rotating the curve y = x³ between y = 0 and y = 4 about the y-axis.

Answer

Exercises

Find the volume generated by the areas bounded by the given curves if they are revolved about the x-axis:

 

(1) y = x, y = 0 and x = 2.

Answer

(2) y = 2x − x² and y = 0 [about x-axis]

Answer

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Find the volume generated by the areas bounded by the given curves if they are revolved about the y-axis:

(3) y² = x, y = 4 and x = 0 [revolved about the y-axis]

Answer

(4) x² + 4y² = 4 (quadrant I) [revolved around y-axis]

Answer

Applications

1. A wine cask has a radius at the top of 30 cm and a radius at the middle of 40 cm. The height of the cask is 1 m. What is the volume of the cask (in L), assuming that the shape of the sides is parabolic?

Answer

Let's see this in LiveMath:

LIVEMath

 

2. A watermelon has an ellipsoidal shape with major axis 28 cm and minor axis 25 cm. Find its volume.

Historical Approach: Before calculus, one way of approximating the volume would be to slice the watermelon (say in 2 cm thick slices) and add up the volumes of each slice using V = πr²h.

Interestingly, Archimedes (the "Eureka! I've got it" dude) used this approach to find volumes of spheres around 200 BC. The technique was almost forgotten until the early 1700s when calculus was developed by Newton and Leibniz.

We see how to do the problem using both approaches.

Volume using historical method:

Answer

"Exact" Volume (using Integration):

Answer

 

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Applications of Integration
• 1. Applications of the Indefinite Integral
• 2. Area Under a Curve
• 3. Area Between 2 Curves
• 4. Volume of Solid of Revolution
• 5. Centroid of an Area
• 6. Moments of Inertia
• 7. Work by a Variable Force
• 8. Electric Charges
• 9. Average Value
• Head Injury Criterion
• 10. Force by Liquid Pressure

• Comments, Questions?

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