Problem

Each integral represents the volume of a solid. D…

00:42

Problem 30 Medium Difficulty

Each integral represents the volume of a solid. Describe the solid.

$ \displaystyle \int_{1}^3 2 \pi y \ln y dy $

Answer

$f\left(y_{1}=\ln y \quad\right.$ or $\quad y=In \quad y=e^{x}$

Discussion

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Video Transcript

in this problem were given an integral and we're also told that this integral represents a volume of a solid. So the volume of revolution. So the first thing that we noticed is that everything is in terms of why? And that would remind us of the formula the volume of revolution. But a curve from what I will say to why it would be, it's actually equals to buy times F F Y squared, I'm steve Y. So this form is quite reminiscent of the volume formula. So we can actually tweak it a little bit so we can factor at the by and that will give us mhm. Do Y. James L N. Y. DY. Now that means that our curve F F Y. The whole squared Is equal to two Y. Times the natural log of Why? Yeah, no, we can take the square root on both sides. But what I like to do first is transfer this to using the Yes mm fire property, flog rhythms. Actually let's just leave it here and take the square root. This is that gives us F. F. Y equals The Square Root of two. Y. Times The Natural Log of Y. We're going to get my hands. And this curve is being rotated from y equals one to y equals three. And that means are required solid. Is is the curve F. F Y equals the square root of two. Y. Times the natural log of Y rotated. You're right uh about the Y axis. Pretty. Do they really want yeah. Yeah. From why equals one To Y. is equal to three. And that is a required solid.

Umar Sohail Q.

Vanderbilt University

Topics

Applications of Integration

Calculus: Early Transcendentals

Chapter 6

Applications of Integration

Section 3

Volumes by Cylindrical Shells

Problem