Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines.

Question

y = x²

y = 6x − x²

a. the x axis

b, y=10

Answer 1

By graphing, we can see that the curves intersect at (0,0) and again at (3,9). The concave down parabola, y = 6x - x², is above the other.

Since both given lines are horizontal, we can use volumes by disks, integrating with respect to x:

a. Vol = π · ∫₀³ (6x - x²)² - (x²)² dx

b. Vol = π · ∫₀³ (10 - x²)² - (10 - (6x - x²))² dx