Problem 25 - University Calculus: Early Transcendentals

Question

Sketch the region of integration and convert each polar integral or sum of integrals to a Cartesian integral or sum of integrals. Do not evaluate the integrals. $$\int_{0}^{\pi / 4} \int_{0}^{2 \sec \theta} r^{5} \sin ^{2} \theta d r d \theta$$

Step-by-Step Solution

Answer

The polar integral converts to \( \int_{0}^{2} \int_{0}^{x} (x^2 + y^2)^{3/2} y^2 \, dy \, dx \) in Cartesian coordinates.

Step 1: Understand the Polar Integral

The given integral is \( \int_{0}^{\pi / 4} \int_{0}^{2 \sec \theta} r^{5} \sin ^{2} \theta \, dr \, d \theta \). This represents a region in polar coordinates where \( r \) varies from 0 to \( 2 \sec \theta \) and \( \theta \) varies from 0 to \( \frac{\pi}{4} \).

Step 2: Sketch the Region of Integration

In polar coordinates, \( r = 2 \sec \theta \) is a line that, when converted to Cartesian coordinates, becomes the line \( x = 2 \). The angle \( \theta = 0 \) represents the positive x-axis, while \( \theta = \frac{\pi}{4} \) represents the line \( y = x \). Thus, the region is a sector of a circle with radius 2 stretching from the x-axis to the line \( y = x \).

Step 3: Derive the Cartesian Conversion Formulas

Use the conversion formulas from polar to Cartesian coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). This gives bounds for \( x \) from \( 0 \) to \( 2 \) and \( y \) from \( 0 \) to \( x \).

Step 4: Express the Integral in Terms of Cartesian Coordinates

Using the conversion formulas, the original integral becomes: \( \int_{0}^{2} \int_{0}^{x} (x^2 + y^2)^{5/2} \frac{y^2}{x^2 + y^2} \, dy \, dx \). This accounts for the transformation from polar to Cartesian coordinates, and the integrand reflects the relationship \( r = \sqrt{x^2 + y^2} \).

Step 5: Set Up the Cartesian Integral

The Cartesian integral is now set up in terms of \( x \) and \( y \) coordinates: \[ \int_{0}^{2} \int_{0}^{x} \left((x^2 + y^2)^{5/2} \cdot \frac{y^2}{x^2 + y^2}\right) \, dy \, dx \]. This represents the Cartesian equivalent of the given polar integral over the described region.

Key Concepts

Polar Coordinates

Polar coordinates offer a way to represent points on a plane using a distance and an angle. Instead of the typical Cartesian grid, which describes a point by its horizontal and vertical position, polar coordinates use:

• Radius (\(r\)): This is the distance from the origin to the point.
• Angle (\(\theta\)): This determines the direction of the point from the origin, measured from the positive x-axis.

In the context of integration, polar coordinates can simplify the description of circular and angular regions. In our exercise, the region is described using polar coordinates ranging from \(0 \leq r \leq 2 \sec \theta\) and \(0 \leq \theta \leq \frac{\pi}{4}\).
This representation aligns neatly with circular boundaries, making polar coordinates ideal for sectors of circles.

Cartesian Coordinates

Cartesian coordinates are the most common system for locating points. They define a point with two numbers, \(x\) and \(y\):

• \(x\): This is the horizontal distance from the origin.
• \(y\): This is the vertical distance from the origin.

When dealing with integrals and transformations, converting polar equations to Cartesian form can make subsequent steps easier. In our exercise, the line \(r = 2 \sec \theta\) converts to \(x = 2\)in Cartesian coordinates. This identifies a vertical line at \(x = 2\), limiting the region of integration on the right.

Integration Bounds

Integration bounds define the limits within which integration is performed. In polar coordinates, these are given in terms of \(r\) (radius) and \(\theta\) (angle). For instance, in our exercise, \(r\) ranges from \(0\) to \(2 \sec \theta\), while \(\theta\) spans from \(0\) to \(\frac{\pi}{4}\).
Once converted to Cartesian coordinates, these bounds change to simple numerical limits:

• \(x\) goes from\(0\) to \(2\).
• \(y\) goes from\(0\) to \(x\).

This transformation adapts the region boundaries using Cartesian grid references, simplifying the process of setting up the double integral.

Coordinate Transformation

Coordinate transformations are essential for converting problems from one coordinate system to another. For our integral, transforming from polar to Cartesian coordinates involves replacing \(x\) and \(y\) for each \(r\) and \(\theta\) using the formulas:

• \(x = r \cos \theta\).
• \(y = r \sin \theta\).

Such transformations adapt mathematical expressions to new coordinate systems. In the problem discussed, we've transformed a polar integral to its Cartesian form, making the integrals more straightforward to evaluate visually and computationally.
This step is crucial for translating circular or angular areas into rectangular coordinates, aligning the problem-solving process with Cartesian toolkits.