91Ó°ÊÓ

Integration in polar coordinates allows us to calculate areas in a way that's perfect for curves described with the polar coordinates system. Instead of using rectangles, as is standard in Cartesian coordinates, polar coordinates divide regions into sectors, like slices of pie. When integrating in polar coordinates, an area element is represented by a tiny sector of the circle. The area of this sector can be approximated as \( \frac{1}{2} r^2 \, d\theta \), where \( r \) is the radius, and \( d\theta \) is the small angle in radians.
For integration, you set up an integral that sums these tiny sector areas over a certain range of \( \theta \) values. It's essential to identify the correct limits for \( \theta \) to ensure you cover the entire region of interest.

• Finding Limits: Determine where your curve starts and ends within the region. In our example, these angles were \( -\frac{\pi}{4} \) to \( \frac{\pi}{4} \).
• Formulating the Integral: If the expression for \( r \) is simple, substitution into the integral is straightforward, as seen with \( r = 2 \).

Remember, the integral essentially calculates the sum of all these small sector areas to find the total area.

Finding area in polar coordinates involves integrating over a range of angles. In our example, we calculated the area of a region inside a circle and to the right of a line.
The integral was set up to find the common region, from the angle \( -\frac{\pi}{4} \) to \( \frac{\pi}{4} \). The formula used was: \[ A = \int_{\theta_1}^{\theta_2} \frac{1}{2} r^2 \, d\theta \]When \( r \) is a constant value (e.g., a circle with a fixed radius), it simplifies the integral, as seen when \( r = 2 \) making the integrand constant.

• Substitution: Directly substitute the value for \( r \) into the integral.
• Calculating: Perform the integration, realizing that \( \theta \) simply scales the area found by half the radius squared.

For our problem, the evaluation of the integral resulted in the total area as \( \pi \), which is quite elegant when calculations are correct.

To find the intersection of polar curves, set their equations equal to each other and solve for \( \theta \). This approach provides the points where the curves meet. For the provided exercise, the circle \( r = 2 \) and the line \( r = \sqrt{2} \sec \theta \) intersect at angles \( \theta = \frac{\pi}{4} \) and \( \theta = -\frac{\pi}{4} \).

• Steps to Find Intersection: Equate their \( r \) values and solve the resulting trigonometric equation.
• Analyzing Solutions: Determine which \( \theta \) values are valid for the original problem. It often helps to visualize the problem or sketch the curves.

Understanding intersections is crucial, as they define the limits of integration and thus ensure accurate area calculation. Working through these intersections first helps to solve the larger problem effectively.