Question

The base of a solid is the region bounded by the lines y = 5x, y = 10, and x = 0. Answer the following.

a) Find the volume if the solid has cross sections perpendicular to the y-axis that are semicircles.

b) Find the volume if the solid has cross sections perpendicular to the x-axis that are semicircles.

I'm confused as to how to do this. Could you please help? Thanks!

Answer 1

(a) The semicircles have a diameter equal to x, or y/5. So, each semicircle has an area of

πd^2/2 = π/2 (y/5)^2 = πy^2/50

Now add up all the thin discs and you get a volume of

∫[0,10] π/50 y^2 dy = 20π/3

(b) Now the sections have diameter equal to y = 5x, so their area is

π/2 (5x)^2 = 25π/2 x^2

and thus the volume is

∫[0,2] 25π/2 x^2 dx = 100π/3

Answer 2

Thank you so much!