Step 3: The formula for the method of shells says that if you have a region between two
curves, f (x) and g(x) from x = a to x = b, then the volume generated when the region is
revolved around the y-axis is: 2π x [f(x) − g(x)] dx if f(x is above g(x) throughout the region.
Thus, our integral is
2π x[(9 − x^2 ) − (9 − 3x)] dx
We can simplify this integral to 2π x(3x − x^2 ) dx = 2π (3x^2 − x^3 ) dx.
Step 4: Evaluate the integral.
- A This problem requires you to be familiar with the Mean Value Theorem for Integrals, which we
use to find the average value of a function.
Step 1: If you want to find the average value of f (x) on an interval [a, b], you need to evaluate
the integral f(x) dx. So here we evaluate the integral ln^2 x dx.
You have to do this integral on your calculator because you do not know how to evaluate this
integral analytically unless you are very good with integration by parts!
Use fnint. Divide this by 2 and you will get 1.204.
- D This problem is testing your knowledge of the Second Fundamental Theorem of Calculus. The
theorem states that f (t) dt = f(u) , where a is a constant and u is a function of x. So
all we have to do is follow the theorem: cos(t) dt = 3cos 3x
- B In order to find the average value of a function, f, on the interval [a, b], we need to evaluate
f(x)dx. Here, we get: 4cos(2x)dx.
