### Home > CALC > Chapter 8 > Lesson 8.3.1 > Problem 8-101

The graph at right is *f*(*x*) = *x *cos *x* + 4 for 0 ≤ *x *≤ 5. Write an integral that could compute the volume of the solid when this region is rotated about:

The

*x*-axis.The

*y*-axis.The line

*x*= 6.The line

*y*= −7.

Vertical rectangles.

Horizontal rectangles would be very difficult (because the inverse of *f*(*x*) is NOT a function). So use the Shell Method.

By shells:

Where did the (6 − *x*) come from?

Recall that each cylindrical shell is a prism with dimensions: (base)(height)(width) = (2π*r*)(*f*(*x*))(*dx*).

Well, (6 − *x*) represents the *r* (radius) in the circumference part of the formula: *C* = 2π*r*.

If we were rotating about the *y*-axis (as in part (b)), then the radius would be the same as the bounds, *x*.

But since we are rotating about a line that is parallel to the *y*-axis, we do not want the radius to be the same as the bounds.

When we plug in the lower bound (*x* = 0), we actually want a radius that is 6.

And when we plug in the upper bound (*x* = 5), we actually want a radius that is 1.

(6 − *x*) will make this work!

Vertical rectangles with a hole in the center.