Question

Use the shell method to write and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis. x + y² = 16

Answer 1

To find the volume of the solid generated by revolving the plane region about the x-axis, we can use the shell method.

Step-by-step explanation:

To find the volume of the solid generated by revolving the plane region about the x-axis, we can use the shell method. First, we need to rewrite the equation in terms of y, so we have y = sqrt(16 - x). The region is bounded by the y-axis, so the limits of integration will be from 0 to the y-coordinate where the curve intersects the y-axis (which is 4).

Now, we can use the formula for the volume of a solid generated by revolving a curve about the x-axis using shells: V = 2π ∫(x * f(x)) dx, where f(x) is the radius of each shell. In this case, f(x) = sqrt(16 - x).

By plugging in the limits of integration and evaluating the integral, we can find the volume of the solid.