# Lesson 2Any Way You Spin ItDevelop 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 -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 -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 -axis? Draw and describe the solid of revolution formed by rotating this semicircle about the -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 -axis? Draw and describe the solid of revolution formed by rotating this triangle about the -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 -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 , , and .

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 -axis to generate it.

### 12.

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

Sketch a graph of the exponential function on the interval . Draw the solid of revolution formed when this piece of the function is revolved about the -axis.

Calculate the area of each of the circular cross sections of the solid of revolution for each integer position along the ­-axis.

## Takeaways

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

## 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 to the nearest of a degree.

### 2.

Use the formula to find the volume of the right triangular prism.