Page 65 - Calculus for the Clueless, Calc II

The integrals are almost always easy. Once you understand the picture, all will be easy. But it takes most people
time, studying the pictures.

Let us get back to our apple. Suppose we core it. When we take slices perpendicular to the axis, we get a ring.
The area of a ring is the area of the outside minus the area of the inside.

The volume of each disc is h. Again, Again, h is small.

Example 6—

Find the volume if our region is rotated about the y-axis.

As you rotate this region, there is a hole. Outside radius is always 9 and the inside radius is always the x value.
2
2
2
But x = y . r = x = y .
4

Example 7—

Find the volume of our glorious region if it is rotated about the line x = 9.

Notice that when we rotate the region about x = 9, there is no hole. V sect = πr h. r = 9 - x = 9 - y . r = 81 -18y +
2
2
2
2
y . h = ∆y.
4