Step 2: (x − 1)^2 dx = (x^2 − 2x + 1) dx
- B Since you are not told which method to use to find the volume you must decide, a big hint is the
answer choices. However, if you didn’t have this hint, then you can use the rule of thumb that it
is TYPICALLY (but not always) better to use cylindrical shells, if the region is bound by more
than two curves (including an axis) or if one or more curves are given as y = and the others are
given as x =. Both conditions are satisfied in this problem, so cylindrical shells is probably
best. The general formula for cylindrical shells is 2π x[f(x) − g (x)] dx. First, the points of
intersection between all these curves must be found, where the region is bound, to establish the
limits of integration. The bounds are x = 5 and x = 10. Next, determine which curve is “on top”
or “more positive.” In this case, the curves in question are y = (x − 5)^3 and y = 0. Since y = (x
− 5)^3 is always more positive than y = 0, y = (x − 5)^3 = f(x) and y = 0 = g(x). Finally, the
general formula is for a region that is rotated about the y-axis, or x = 0. Since our curve is
shifted to be rotated around x = 2, the radius of the cylinder, x, is now x − 2 to account for the
shift. Thus, the final integral is: 2π (x − 2)(x − 5)^3 dx.
- B Use u-substitution in which u = x^3 − 3 and du = 3x^2 dx. Thus, the integral is:
- B In order to determine the equation for the normal line, take the derivative with respect to x at
