Use the general slicing method to find the volume of the solid whose base is the triangle with vertices (0 comma 0 ), (15 comma 0 ), and (0 comma 15 )and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles

volume V of the solid

Step-by-step explanation:

The situation is depicted in the picture attached

(see picture)

First, we divide the segment [0, 5] on the X-axis into n equal parts of length 5/n each

[0, 5/n], [5/n, 2(5/n)], [2(5/n), 3(5/n)],..., [(n-1)(5/n), 5]

Now, we slice our solid into n slices.  

Each slice is a quarter of cylinder 5/n thick and has a radius of  

-k(5/n) + 5  for each k = 1,2,..., n (see picture)

So the volume of each slice is  

for k=1,2,..., n

We then add up the volumes of all these slices

Notice that the last term of the sum vanishes. After making up the expression a little, we get

But

we also know that

and

so we have, after replacing and simplifying, the sum of the slices equals

Now we take the limit when n tends to infinite (the slices get thinner and thinner)

and the volume V of our solid is