Final answer:
To find the volume of the solid formed by rotating the region between two curves about the y-axis, we can use the method of cylindrical shells. The volume can be found by integrating the difference between the upper and lower curves over the interval defined by the x-values where the two curves intersect.
Step-by-step explanation:
To find the volume of the solid formed by rotating the region bounded by the curves y = -x² + 14x - 45 and y = 0 about the y-axis, we can use the method of cylindrical shells.
First, we need to find the limits of integration by setting the two curves equal to each other and solving for x. This gives us -x² + 14x - 45 = 0, which factors as (x - 5)(x - 9) = 0. So the limits of integration are x = 5 and x = 9.
The formula for the volume using cylindrical shells is V = 2π ∫[a,b] x f(x) dx, where f(x) is the height of the shell at x. In this case, f(x) is the difference between the upper curve and the lower curve, which is given by f(x) = -x² + 14x - 45. So the volume V is equal to 2π ∫[5,9] x (-x² + 14x - 45) dx.
Integrating this function gives us the volume of the solid: V = 2π ∫[5,9] (-x³ + 14x² - 45x) dx.