Question

Use cylindrical shells to set up the integral to find the volume of the solid generated by revolving the region bounded by the given equations around the given line Skecth the region 6) x=(y−3)² ,x=4 about y=1

Answer 1

To find the volume of the solid, we can use the method of cylindrical shells. Consider an infinitesimally thin shell with radius y and thickness dy. Find the height of the shell by subtracting the x-values of the curves and the circumference of the shell. The volume of the shell is given by dV = 2πy * h * dy. Integrate this expression over the range of y-values to find the total volume.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by the given equations around the given line, we can use the method of cylindrical shells. The key idea is to consider an infinitesimally thin shell with radius y and thickness dy. We then find the height of the shell by subtracting the x-values of the curves: h = (y - (y - 3)^2) = 3 - (y - 3)^2. The circumference of the shell is 2πy. Therefore, the volume of the shell is given by dV = 2πy * h * dy.

To find the total volume, we integrate this expression over the range of y-values that define the region. In this case, the range is from y = 0 to y = 4. So the integral becomes V = ∫(0 to 4) 2πy * (3 - (y - 3)^2) dy. Evaluating this integral will give us the volume of the solid.