In Exercises 23-26, sketch the region of integration, and...

In Exercises $23-26,$ sketch the region of integration, and convert each polar integral or sum of integrals into a Cartesian integral or sum of integrals. Do not evaluate the integrals.
$$\int_{0}^{\pi / 2} \int_{0}^{1} r^{3} \sin \theta \cos \theta d r d \theta$$

Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle (r, ?) instead of the usual Cartesian coordinates (x, y). This system is especially useful for regions with circular symmetry or when dealing with circles, arcs, and sectors.

Cartesian Coordinates

Cartesian coordinates describe points in the plane using x and y values. They are the most common coordinate system for many integration problems and are especially useful when the region of integration or the function’s formulation is better suited to horizontal and vertical boundaries.

Coordinate Transformation

Converting an integral from polar to Cartesian coordinates involves using the relationships x = r cos? and y = r sin?, along with the fact that r² = x² + y². This transformation is crucial when the integral or the region is originally expressed in polar form, and a reformulation in terms of x and y is desired.

Jacobian Determinant

When changing variables in double integrals, the area element transforms according to the Jacobian determinant. In the case of polar to Cartesian conversion, the differential area element in polar form, r dr d?, appears as the product of the Jacobian (r) and the differentials of the polar coordinates.

Region of Integration

The region of integration must be carefully translated between coordinate systems. For example, an integration over r from 0 to 1 and ? from 0 to ?/2 in polar coordinates corresponds to a quarter of a unit disk in the first quadrant in Cartesian coordinates, described by the inequalities x² + y² ? 1 with x ? 0 and y ? 0.

Transcript

00:01 We want to sketch the region of integration that we're given here, and then convert each of the polar integrals into either a cartesian integral or a sum of cartesian integrals.

00:13 And they want us to not evaluate the integral.

00:16 All right, so let's go ahead and first figure out how to graph this, or what region we're graphing over.

00:23 So this tells us r is going to vary between 0 and 1, and then our outer bound here tells us that theta should range from 0 to pi half.

00:39 Let's go ahead and first just draw a quick sketch.

00:43 So if it's going from 0 to pi half, let's go ahead and put that down first.

00:46 So we know 0 is here, pi half is here.

00:49 So it should be something trapped within this.

00:52 And then they tell us that our radius should range from 0 to 1.

00:57 Well, so in each of these regions, it should just go from 1 to 1, and then we just go ahead and connect them like that.

01:02 So it's just going to be this circle right here.

01:07 So it's a circle of radius 1, and we know how to describe that.

01:14 That's going to be x squared plus y squared is equal to 1 squared, which is just 1.

01:20 And we can go ahead and solve for x or y.

01:24 So let's just go ahead and solve for y.

01:26 And doing that should give us that y is equal to 1 minus x squared square rooted, just y...

Please give Ace some feedback