Using the method of cylindrical shells, the volume of the solid obtained by rotating the region bounded by the curves xy = 8, x = 0, y = 8, and y = 10 about the x-axis is 400π cubic units.
To find the volume of the solid obtained by rotating the region bounded by the curves xy = 8, x = 0, y = 8, and y = 10 about the x-axis, we can use the method of cylindrical shells. This method involves integrating the circumference of each shell multiplied by its height to obtain the volume.
1. Determine the limits of integration:
- Since the region is bounded by x = 0 and xy = 8, we need to find the x-values where the two curves intersect.
- Solving xy = 8 for x, we get x = 8/y.
- Setting this equal to x = 0, we find y = 0. Therefore, the lower limit of integration is y = 0.
- To find the upper limit of integration, we set xy = 8 equal to y = 10 and solve for x: 10x = 8, x = 8/10 = 0.8. Therefore, the upper limit of integration is x = 0.8.
2. Set up the integral:
- The circumference of each shell is given by 2πr, where r is the distance from the x-axis to the curve xy = 8.
- Since r = y, the circumference is 2πy.
- The height of each shell is given by the difference in y-values, which is y = 10 - 8 = 2.
- Therefore, the integral for the volume is ∫(2πy)(2) dy, integrated from y = 0 to y = 10.
3. Evaluate the integral:
- Integrating ∫(2πy)(2) dy from y = 0 to y = 10 gives the volume V = ∫4πy dy evaluated from 0 to 10.
- Evaluating the integral, we get V = 4π(10^2 - 0^2) = 400π cubic units.