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Problem 66 Hard Difficulty

Find the volume common to two circular cylinders, each with radius $ r $, if the axes of the cylinders intersect at right angles.

Answer

$\frac{16 r^{3}}{3}$

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Video Transcript

okay. Recall the fact that we know that Z is a prevalent to the squirt of r squared minus Z squared. Now remember, we know the whole cross section is gonna be four times that. Remember when we square something, it cancels out the square root and the squared sine so we end up with simply four times r squared minus c squared. Now that we have this, we know we can integrate from zero to our And remember, we can actually just double this. This makes it a lot easier because now, instead of having negative bounds who are simply doubling the result, pull out all of our constants. Other words, if we plot before we end up with 82 times four is eight. Integrate. When we integrate, we use the power method, which means we increased the expert by one, we divide by the new expert. Z squared becomes 1/3 c cubed. Now we're at the point where we can plug in our bounds. Remember, we're playing and are on the top and they're playing in zero on the bottom. We plug in zero. We actually just end up with zero minus zero, which is always just zero, which means we have eight times to r cubed over three, which means we end up with 16 r cubed divided by three.