In Exercises 45-50, use the most convenient method (Disk or Shell Method) to find the given volu

In Exercises 45-50, use the most convenient method (Disk or...

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Key Concepts

Shell Method

The shell method computes the volume of a rotated region by dividing it into thin cylindrical shells. These shells are parallel to the axis of rotation, and their volumes are computed by integrating the lateral surface area of each shell multiplied by its thickness, which usually simplifies the integral in cases where the disk/washer method is cumbersome.

Definite Integration

Definite integration is the process of summing an infinite number of infinitesimally small quantities to calculate a total amount, such as area or volume. In the context of volumes of revolution, it is used to add up the contributions of all the slices or shells over a specified interval.

Volume of Revolution

This concept involves finding the volume of a solid created when a region in the plane is rotated about an axis. It requires setting up and evaluating an integral that accumulates the volume of infinitesimally thin slices or shells formed by the rotation.

Disk/Washer Method

The disk/washer method is a technique for computing volumes of revolution by slicing the solid perpendicular to the axis of rotation. When the slices are taken perpendicular to the axis, they form disks or washers. The volume is then found by integrating the area of these circular slices along the axis of rotation.

Transcript

00:01 We're asked to find the volume of tank by rotating the region enclosed by these two functions about the y -axis.

00:08 So let's draw a region first and what the solid is going to look like.

00:13 So this is going to be a sideways parabola.

00:17 Let's say that's five.

00:20 It's going to look like this.

00:26 So the region that we're revolving is this one right here.

00:32 So when we revolve that about the y -axis, we're going to end up with something.

00:36 That looks like this.

00:42 So if we were to use the shell method, it would be vertical cylinders, so which means we are going to be integrating in the direction of x.

00:53 But since we're already given the function as x in terms of y, that tells us we should be using the disk method, then we don't have to, we don't have to change this function.

01:02 Because if we use the disk method, we would be integrating in a direction of y.

01:08 So let's write down the wall.

01:09 Formula for using the disk method that is going to be equal to pi integral of radius squared.

01:20 Okay, so let's draw one of our disks.

01:25 Let's say here.

01:28 So one of our disks would look like this.

01:35 This would be one of our disks.

01:38 So then the radius of our disk, that is just the x value.

01:44 So which is the function.

01:46 Here...