Problem 26 Easy Difficulty

Cylindrical shells In Section $6.2,$ we learned how to find the volume of a solid of revolution using the shell method; namely, if the region between the curve $y=f(x)$ and the $x$ -axis from $a$
to $b(0 < a < b)$ is revolved about the $y$ -axis, the volume of the resulting solid is $\int_{a}^{b} 2 \pi x f(x) d x .$ Prove that finding volumes by using triple integrals gives the same result. Use cylindrical
coordinates with the roles of $y$ and $z$ changed.)

Answer

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Video Transcript

so the proof. This is the start with a substitution for cylindrical coordinates. So for cylindrical, we have X rays or comes on theater. Why's what Selvan Z is for Cynthia? That's over. Substitution for cylindrical coordinates. So hence the bull. We'll be equals shoe. So we're gonna expressed of all, um, in cylindrical hornets. Witches are the hors de theatre B Y. So you know, we're setting which just up flogging the values. This is a to B 0 to 2 x and zeros out of our or the white, the state of the or Okay, So the evaluation of this since the girl is finally to buy some say to be or f or the or, huh? So what is the same thing as to part from a to B X of That's the ends. Still, that shouldn't coordinates. We can prove surgical internal will be transformed into a simple one. Variable. Devon's into group. That's it.