Lesson 4 Any Way You Spin It Develop Understanding

Learning Focus

Develop a strategy for drawing solids of revolution.

The blur of a spinning penny takes on the visual shape of a sphere. What might a spinning triangle, rectangle, or trapezoid look like?

How would the shape created by the spinning object change if the object is rotated about a different axis?

Open Up the Math: Launch, Explore, Discuss

You might have played with a spinning top, used a pottery wheel, or watched a figure skater spin so rapidly they looked like a solid blur. The clay bowl, the rotating top, and the spinning skater—each of these can be modeled as solids of revolution—a 3-D object formed by spinning a 2-D figure about an axis.

Suppose the right triangle is rotating rapidly about the $x$ -axis. Like the spinning skater, a solid image would be formed by the blur of the rotating triangle.

1.

Draw and describe the solid of revolution formed by rotating this triangle about the $x$ -axis.

2.

Find the volume of the solid formed.

3.

Describe your strategy for drawing the solid of revolution for this triangle.

4.

What would the figure look like if the semicircle rotates rapidly about the $x$ -axis? Draw and describe the solid of revolution formed by rotating this semicircle about the $x$ -axis.

5.

Find the volume of the solid formed in problem 4.

6.

What would the figure look like if the triangle rotates rapidly about the $y$ -axis? Draw and describe the solid of revolution formed by rotating this triangle about the $y$ -axis.

7.

Find the volume of the solid formed in problem 6.

8.

What about the following 2-D figure? Draw and describe the solid of revolution formed by rotating this figure about the $x$ -axis.

9.

Draw a cross-section of the solid of revolution formed by the figure in problem 8 if the plane cutting the solid is the plane containing the coordinate axes.

10.

Draw some cross-sections of the solid of revolution formed by the figure in problem 9 if the planes cutting the solid are perpendicular to the plane containing the coordinate axes. Draw the cross-sections when the intersecting planes are located at $x = 5$ , $x = 10$ , and $x = 15$ .

So, why are we interested in solids that don’t really exist—after all, they are nothing more than a blur that forms an image of a solid in our imagination. Solids of revolution are used to create mathematical models of real solids by describing the solid in terms of the 2-D shape that generates it.

11.

For each of the following solids, draw the 2-D shape that would be revolved about the $x$ -axis to generate it.

a.

b.

c.

12.

What issues arise when modeling these objects as solids of revolution?

Ready for More?

Sketch a graph of the exponential function $f (x) = 2^{x}$ on the interval $- 2 \leq x \leq 2$ . Draw the solid of revolution formed when this piece of the function is revolved about the $x$ -axis.

Calculate the area of each of the circular cross sections of the solid of revolution for each integer position $- 2 \leq x \leq 2$ along the $x$ -axis.

Takeaways

Strategies for drawing, describing, or analyzing a solid of revolution:

Vocabulary

disc or disk
solids of revolution
washer
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to create solids of revolution by rotating a 2-D shape around an axis of rotation and we examined the cross-sections that are formed when solids of revolution are sliced perpendicularly to the plane that contains the axes.

Retrieval

1.

Use the given measures on the triangle to write the indicated trigonometric value. Then find the measure of angle $A$ to the nearest $100 th$ of a degree.

$\sin A = \underset{―}{}$

$\cos A = \underset{―}{}$

$\tan A = \underset{―}{}$

$m ∠ A = \underset{―}{}$

2.

Use the formula $V = B h$ to find the volume of the right triangular prism.