Question

The base of a solid is the region in the xy-plane bounded by y=3x² and y=12. Cross-sections perpendicular to the y-axis are isosceles right triangles with one leg in the base. Find an integral for the volume of the solid DO NOT EVALUATE.

Answer 1

To find the volume of the solid with isosceles right triangle cross-sections perpendicular to the y-axis, we need to integrate the area of each cross-section over the height of the solid.

Step-by-step explanation:

To find the volume of the solid, we need to determine the area of each cross-section and integrate it over the height of the solid.

Given that the cross-sections are isosceles right triangles, one leg will be the base of the solid, which is the region bounded by y = 3x² and y = 12 in the xy-plane.

Let's denote the length of this base as b, and the height of the solid as h. Then, the area of each cross-section will be 1/2 * b * b = (1/2) * b², since it's an isosceles right triangle. The volume of the entire solid will be the integral of (1/2) * b² with respect to the height h.