Question

Find the volume V of the solid obtained by rotating the region bounded by the curve y=x^2+1, x=0, and y=5 about the x-axis. (Express numbers in exact form. Use symbolic notat)

Answer 1

To find the volume of the solid obtained by rotating the region bounded by the curve y=x^2+1, x=0, and y=5 about the x-axis, we can use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the curve y=x^2+1, x=0, and y=5 about the x-axis, we can use the method of cylindrical shells.

Each shell has a radius equal to the x-coordinate of the curve (x), a height equal to the difference in y-coordinates of the curve (y=5-(x^2+1)),

and a thickness characterized by dx. The volume of each shell is therefore given by dV=2πx(5-(x^2+1))dx.

To find the total volume, we integrate the expression for dV from x=0 to x=√4, since these are the x-coordinates of the curve.

The integral is given by V=∫[0 to √4] 2πx(5-(x^2+1))dx. Evaluating this integral gives the exact volume of the solid.