08.06 Volumes with Discs

The function y = (x + 3)2 on the domain [−3, ∞), is bounded by the vertical line x = −3 and the horizontal line y = 9. Which of the following gives the volume of the solid generated when the region is revolved around the vertical line x = −3?

127.2345

Let R be the region in the first quadrant bounded by x2 − y2 = 16, the horizontal line y = 0, and the vertical line x = 5. What is the volume of the solid formed by revolving R about the x-axis?

13.6136

A region in the first quadrant is bounded by the graph y equals two thirds times x, the y-axis, and the horizontal line y = 4. What is the volume of the solid generated when this region is rotated around the y-axis?

48π

The curve f(x) = cos(x) + 2 from x = 0 to 2π is rotated about the x-axis to form the shape of an hourglass. If x and f(x) are measured in centimeters, then what is the volume of the hourglass in cubic centimeters?

88.8264

A region in Quadrant II is bounded by the graph y = (x + 3)2 and the horizontal line y = 9. Which of the following gives the volume of the solid generated when the region is revolved around the horizontal line y = 9?

V equals pi times the integral of the quantity 9 minus the quantity x plus 3 end quantity squared end quantity squared dx from negative 6 to 0