Lesson 2 Any Way You Spin It Develop Understanding
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
Draw and describe the solid of revolution formed by rotating this triangle about the
Find the volume of the solid formed.
Describe your strategy for drawing the solid of revolution for this triangle.
What would the figure look like if the semicircle rotates rapidly about the
Find the volume of the solid formed in problem 4.
What would the figure look like if the triangle rotates rapidly about the
Find the volume of the solid formed in problem 6.
What about the following 2-D figure? Draw and describe the solid of revolution formed by rotating this figure about the
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.
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
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.
For each of the following solids, draw the 2-D shape that would be revolved about the
What issues arise when modeling these objects as solids of revolution?
Ready for More?
Sketch a graph of the exponential function
Calculate the area of each of the circular cross sections of the solid of revolution for each integer position
Strategies for drawing, describing, or analyzing a solid of revolution:
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.
Use the given measures on the triangle to write the indicated trigonometric value. Then find the measure of angle
Use the formula