Improper integrals and volumes of revolution

What you'll learn

1. 1

   Imagine a bucket being filled with paint. The bucket has a curved shape — like a vase — and as the paint pours in, the surface rises. The volume of paint is the area under the curve, rotated around!

2. 2

   Here's a simple curve y = x² from x=0 to x=2. When spun around the x-axis, it makes a solid like a trumpet bell!

3. 3

   Let's find the volume when y = x² from x=0 to x=2 is rotated around the x-axis.

4. 4

   Spin the curve y = √x from x=0 to x=4 around the x-axis. Drag the slider to see the solid form!

5. 5

   What is the formula for the volume of a solid of revolution around the x-axis?

6. 6

   Now try an improper integral: volume of y = 1/x from x=1 to ∞ rotated around x-axis.

7. 7

   When evaluating an improper integral to ∞, what must the result be for the volume to exist?