Question

Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis. x=3+y

2 ,x=0,y=1,y=2

Answer 1

To calculate the volume of the solid formed by rotating a given region around the x-axis using the method of cylindrical shells, we set up an integral representing the sum of the volumes of infinitesimally thin shells over the interval y=1 to y=2 and evaluate it.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves x=3+y², x=0, y=1, and y=2 about the x-axis using the method of cylindrical shells, we consider a representative shell at a distance y from the x-axis with a height of x=3+y² (from x=0 to x=3+y²) and thickness dy. The volume dV of this shell is the circumference of the shell (2πy) times its height (3+y²) times the thickness (dy).

Therefore, the volume of the shell is dV=2πy(3+y²)dy. To find the total volume, we integrate this expression from y=1 to y=2.

V = ∫_{1}^{2} 2πy(3+y²)dy

By evaluating this integral, we can determine the volume of the solid.