Question

the base of the solid is the triangle enclosed by =7, the ‑axis, and the ‑axis. the cross sections perpendicular to the ‑axis are semicircles. compute the volume of the solid.

Answer 1

To find the volume of a solid with semicircular cross sections, calculate the integral of the area of the cross sections along the length of the base of the triangle from x = 0 to x = 7.

Step-by-step explanation:

To calculate the volume of a solid with semicircular cross sections perpendicular to the x-axis, where the base is a triangle in the xy-plane with vertices at (0,0), (7,0), and (0,7), we can use integral calculus. The area of a semicircle with radius r is given by ½πr². Since the triangle lies in the first quadrant, we can express y as a linear function of x, specifically, y = 7 - x. For each value of x, the radius of the semicircle is y/2, so the area of the cross section is ½π(½(7 - x))². Integrating this area function from x = 0 to x = 7 gives us the volume of the solid.

The integral is ½π∫ (½(7 - x))² dx from x = 0 to x = 7. Carrying out this integral will yield the desired volume.