E1 D please answer this question step by step thanks Find th

E1 D please answer this question step by step. thanks

Find the volume of the solid that is bounded by two right circular cylinders of radius r, if their axis meet at angle theta.

Solution

Imagine that the cylinders have the x and y axes as their respective axes of symmetry. Then we can obtain them by the equations

x² + z² = r² (has y-axis as its axis of symmetry)

y² + z² = r² (has x-axis as its axis of symmetry)

If you draw even a marginal sketch, you see that they cross with the origin right in the center. There is a lot of symmetry here, so that means we can consider only the first octant, find the volume of that and multiply through by 8. I\'m actually going to consider one half of the first octant part and multiply by 16. If you cut the intersection with the planes y = x and y = - x, you can consider only the 16th of the solid in the first octant bounded on one side by the yz-plane, below by the xy-plane, and by the plane y = x.

If you cut this thing in half and looked at how it intersects the xy-plane, the cross section in the xy-plane looks like a square with corners at (r,r), (-r,r), (r,-r) and (-r,-r). Since we are interested only in the first octant, we can focus on the first quadrant part. Consider the triangle in the xy-plane with bounds

0 y x, 0 x r.

The part of the solid above this is x² + z² = r². This gives z bounds

0 z (r² - x²).

This gives all of the limits of integration. The volume is

...........r.x.(r² - x²)
V = 16 dz dy dx
..........000

It\'s not tough to integrate, because after integrating out z and y, the integrand is x(r² - x²) which can be done using a simple substitution. The volume ends up being V = (16/3) r^3