Find the volume of the solid obtained by rotating the region enclosed by the curves y = 15-x, y = 3x+11 and x = -1 about the x-axis.

Answer 1

First, graph the equations so you can see what the region looks like. In this case, the region is a triangle, and the line y=15-x is above y=3x+11.

Now the integral equation for a volume rotated around x using the washer method is

V = ∫π(OR²-IR²)dx with bounds a to b

OR stands for outer radius and IR stands for inner radius. In this case, OR is the line 15-x and IR is the line 3x+11. So plugging them in you get

V = π∫(15-x)²-(3x+11)² dx

To find the bounds, you have to figure out the left most intersection point and the right most intersection point. You can do this by setting the three equations equal to each other two at a time. The left bound will be -1 and the right bound corresponds to the x-coordinate of the intersection point (1,14), so 1.

I will let you work the integration steps out yourself, but your answer should be about 636.7 units cubed.