Question

Find the volume of the solid whose base is the region in the first quadrant bounded by y=x⁶, y=1, and the y axis and whose cross-sections perpendicular to the x axis are semicircle.

Answer 1

To find the volume of the solid, we can use the method of cross-sectional areas. The area of each semicircle can be found by using the equation of the curve to define the radius. Integrating the area of each semicircle will give us the volume of the solid.

Step-by-step explanation:

To find the volume of the solid, we can use the method of cross-sectional areas. Since the cross-sections perpendicular to the x-axis are semicircles, we can find the area of each semicircle and integrate it along the x-axis to find the volume.

The radius of each semicircle is given by y = x^6, so the area of each semicircle is A(x) = (1/2) * pi * (x^6)^2. Integrating this from x = 0 to x = 1 will give us the volume of the solid.

Therefore, V = ∫10 (1/2) * pi * x^12 dx = (1/14) * pi.