The graph at right is $f(x) =\frac { 1 } { 2 }x\operatorname{cos} x + 4$ for $0 ≤ x ≤ 5$. Write an integral that could compute the volume of the solid when this region is rotated about:

1. The $x$-axis.

   Hint (a):

   Vertical rectangles.

2. The $y$-axis.

   Hint (b):

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

3. The line $x = 6$.

   Answer (c):

   By shells:

   Hint (c):

   Where did the $(6 − x)$ come from?
   Recall that each cylindrical shell is a prism with dimensions: (base)(height)(width) $= (2πr)(f(x))(dx)$.
   Well, $(6 − x)$ represents the $r$ (radius) in the circumference part of the formula: $C = 2πr$.
   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!

4. The line $y = −7$.

   Hint (d):

   Vertical rectangles with a hole in the center.