Find the volume generated by rotating the area bounded by the graphs of each set of equations ar

step 1 So we want to find the volume when we rotate around the y -axis. And our functions are y -equals

Find the volume generated by rotating the area bounded by...

Key Concepts

Cylindrical Shell Method

This method is used to calculate the volume of a solid of revolution, particularly when the region is rotated about an axis parallel to the slices. It involves integrating the lateral area of cylindrical shells, which are formed by a narrow strip of the region when rotated around the axis.

Solids of Revolution

This concept involves generating a three-dimensional solid by rotating a two-dimensional area or curve about an axis. The resulting shape's volume can be computed by integrating over the area, taking into account the rotation which creates circular cross-sections or shells.

Definite Integration

Definite integration is the mathematical process of calculating the accumulation of quantities, such as area or volume, over a specified interval. In the context of solids of revolution, it is used to sum the contributions of individual elements (like shells or disks) across the bounds defined by the intersection of curves.

Transcript

00:01 So we want to find the volume when we rotate around the y -axis.

00:04 And our functions are y -equals x squared, x -equals 0, and x -equals 3.

00:12 Now, i'm interpreting this question.

00:14 I'm not seeing the book.

00:15 I'm interpreting this question as here's our y - equals x -squared, and we're having x -equal -0, so we're only using basically in this first quadrant, and x -equals 3.

00:27 So x -equals 3 would come up here, this and intersect at that point of three and nine.

00:34 And so i believe this is the region that they're discussing.

00:37 I would have more likely had said that y equals zero is the other bounding mark.

00:43 However, i didn't write the book.

00:46 So if we rotate that around the y axis, that's going to come over here.

00:51 And then we can use either a shell method or a disk method.

00:56 And why don't we use the washer method? so if we have one washer, the exterior of the washer is here, and that interior of the washer is here.

01:07 And it's going to have a thickness that is only d .y.

01:10 So we'll have to write the uniferal in terms of y.

01:13 And so we will need this function solved for x, and that would be x equals the square root of y.

01:21 So we can think of any point along here.

01:24 If we take just a random point along here, it has a particular x coordinate of square root.

01:30 Root of y...

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