Find the volumes of the solids generated by revolving the...

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the $y$-axis.
The region enclosed by $x=\sqrt{2 \sin 2 y}, \quad 0 \leq y \leq \pi / 2, \quad x=0$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Solids of Revolution

This concept involves creating a three?dimensional solid by rotating a two-dimensional region around a specified axis. Calculating the volume of these solids is a fundamental application of integration in calculus.

Washer/Disk Method

This method is used to compute the volume of a solid of revolution by slicing the solid perpendicular to the axis of rotation. Each slice forms a disk (or a washer if there is a hole) whose volume can be expressed in terms of the radius and thickness, and the total volume is found by integrating these individual volumes over the given interval.

Setting Up the Integral

This refers to the process of translating the geometric description of a region into an integral expression. It involves identifying the boundaries of the region, expressing the radii in terms of the variable of integration, and determining the appropriate limits of integration to accurately represent the volume generated by the rotation.

Definite Integration

Definite integration is a calculus technique used to compute the exact accumulation of quantities, such as volume, across a given interval. When applied to problems of solids of revolution, it sums the volumes of infinitesimally thin slices (disks or washers) to yield the total volume of the solid.

Transcript

00:01 We want to find the volume of the solid generated by revolving the region.

00:08 X is equal to the square root of 2 times sine 2 y between the interval 0 to pi half on the y axis and x is equal to 0.

00:22 So this here is the graph in question.

00:31 And now what we're going to do is first decide what formula we need to use for this, and it just so happens.

00:41 We would like to use a, b, and i'm going to move the pie out front here.

00:49 We have our radius with respect to y, in this case squared, d, y.

01:00 So let's go ahead and try to figure out what our radius should be.

01:04 So if i just take some random point, since we're rotating about the y -axis, and let me just make sure it actually is the y -axis, yeah, we're rotating about the y -axis.

01:23 What we want to do is go from here to here and then rotate around like this.

01:33 So our radius, in this case, is just going to be how far along until we hit x.

01:45 As you can see, from here to here would really just be the length x, and that would be the square root of two times sine 2 y.

01:58 So we can go ahead and plug that in for r.

02:02 So let's just write it on the sign.

02:04 R is going to be 2.

02:07 Sine 2 .y.

02:19 And now we can go ahead and plug that in.

02:25 So pi...