Question

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.

y=7xy = 21x= 0

Answer 1

The shell method is used to find the volume of a solid generated by revolving a plane region around an axis. In this case, we revolve the region between the curves y=7x and y=21x about the y-axis by setting up and evaluating an integral using the shell method.

Step-by-step explanation:

The shell method is used to find the volume of a solid generated by revolving a plane region around a line. In this case, we are revolving the region between the curves y=7x and y=21x about the y-axis.

1. To set up the integral, we need to determine the bounds of integration. The intersection points of the two curves are x=0 and x=3.
2. We will integrate with respect to y since the region is bounded by y-values. The integral will have the form: V = 2π∫(R(y))(h(y))dy.
3. R(y) represents the distance from the y-axis to the curve, which is given by R(y) = 7x - 0 = 7x. And h(y) represents the height of the cylindrical shell, which is given by h(y) = x2(b) - x1(a) = 3 - 0 = 3.
4. Substituting the equations for R(y) and h(y) into the integral and evaluating it will give us the volume of the solid.