Question

If x≥0, find the volume of the solid obtained by rotating the region enclosed by the graphs about the line y=10.

y=x²,y=6−x,x=0

Answer 1

To find the volume of the solid obtained by rotating the region enclosed by the graphs about the line y=10, use the method of cylindrical shells.

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region enclosed by the graphs about the line y = 10, we can use the method of cylindrical shells.

The region enclosed by the graphs consists of two parts: the area between the graphs of y = x^2 and y = 6 - x, and the area under the graph of y = 6 - x from x = 0 to x = 1.

1. First, find the bounds of integration by setting the two equations equal to each other: x^2 = 6 - x. Solve this quadratic equation to find the x-coordinates of the points of intersection.
2. Next, set up the integral to find the volume using the formula for the volume of a solid of revolution: V = ∫[a,b] 2πy(x)h(x) dx, where y(x) is the height of the shell and h(x) is the length of the shell.
3. Integrate the expression and evaluate the integral to find the volume of the solid.