Let A be the bounded region enclosed by
the graphs of
f(x) = x , g(x) = x3 .
Find the volume of the solid obtained by rotating the region A about the line x + 2 = 0.
To find the volume of the solid obtained by rotating the region A about the line x + 2 = 0, we can use the method of cylindrical shells.
First, let's determine the limits of integration. The line x + 2 = 0 can be rewritten as x = -2, so we need to find the intersection points of the graphs of f(x) = x and g(x) = x^3.
Setting f(x) = g(x), we have x = x^3. Rearranging, we get x^3 - x = 0. Factoring out an x, we have x(x^2 - 1) = 0. This gives us three possible values for x: x = 0, x = -1, and x = 1.
Next, let's set up the integral to find the volume. We'll integrate from x = -1 to x = 1, since these are the bounds of the region A. The height of each cylindrical shell will be the difference between f(x) and g(x), and the radius will be the distance from the line x + 2 = 0 to the point x.
The distance from the line x + 2 = 0 to a point x is simply x + 2. So the radius of each cylindrical shell is r = x + 2.
The height of each cylindrical shell is h = f(x) - g(x) = x - x^3.
The volume of each cylindrical shell can be calculated as V = 2πrh, where r is the radius and h is the height.
Therefore, the volume of the solid obtained by rotating the region A can be found by integrating the volume of each cylindrical shell:
V = ∫[from -1 to 1] 2π(x + 2)(x - x^3) dx
To evaluate this integral, you can expand the expression and integrate each term separately. Once you have the antiderivative, plug in the bounds of integration (-1 and 1) and subtract the result to find the volume.
I hope this explanation helps you understand the process of finding the volume of a solid obtained by rotating a region about a line using cylindrical shells.