Calculating the Area of a Region Defined by 2cos θ: An SEO-Optimized Guide

Understanding the area of geometric regions defined by trigonometric functions is a key skill in calculus and geometric analysis. This guide focuses on the specific case of calculating the area of a region bounded by the polar equation r 2cos θ, where θ varies from 0 to π/2. This topic is particularly relevant for students and professionals in fields such as engineering, physics, and mathematics.

Introduction to the Problem

The polar equation r 2cos θ describes a geometric region in the plane. To find the area enclosed by this region, we can use the concept of double integrals in polar coordinates. This guide will walk you through the detailed process of solving this problem and present the mathematical solution in an SEO-friendly format.

Mathematical Solution

The area (A_k) of the region can be calculated using the following double integral:

[ A_k int_{0}^{pi/2} int_{0}^{k cos theta} r , dr , dtheta ]

This integral can be broken down and solved step-by-step:

First, integrate with respect to (r): [ int_{0}^{k cos theta} r , dr frac{1}{2} left. r^2 right|_{0}^{k cos theta} frac{1}{2} (k cos theta)^2 ]

Next, substitute the result into the outer integral and integrate with respect to (theta): [ A_k frac{1}{2} k^2 int_{0}^{pi/2} cos^2 theta , dtheta ]

Use the trigonometric identity (cos^2 theta frac{1 cos 2theta}{2}) to simplify the integral: [ A_k frac{1}{4} k^2 int_{0}^{pi/2} (1 cos 2theta) , dtheta ]

Separate the integral into two parts: [ A_k frac{1}{4} k^2 left( int_{0}^{pi/2} 1 , dtheta int_{0}^{pi/2} cos 2theta , dtheta right) ]

Evaluate the integrals: [ int_{0}^{pi/2} 1 , dtheta theta Bigg|_{0}^{pi/2} frac{pi}{2} ]

For (int_{0}^{pi/2} cos 2theta , dtheta), use the substitution (u 2theta), hence (du 2dtheta): [ int_{0}^{pi/2} cos 2theta , dtheta frac{1}{2} int_{0}^{pi} cos u , du frac{1}{2} left. sin u right|_{0}^{pi} 0 ]

Combine the results: [ A_k frac{1}{4} k^2 left( frac{pi}{2} 0 right) frac{pi}{8} k^2 ]

In your case, (k 2), so the area is: [ A_2 frac{pi}{8} (2)^2 frac{pi}{2} ]

Conclusion

This guide has provided a detailed step-by-step solution to calculating the area of a region defined by the polar equation r 2cos θ, where θ varies from 0 to π/2. This problem is a classic example in integral calculus and demonstrates the power of using polar coordinates to solve geometric problems.

Further Reading

For more information, you can explore related topics such as:

Area of Polar Regions

Integration in Calculus

Trigonometric Identities

Understanding the methods used in this problem will greatly enhance your skills in handling similar questions in geometry and calculus.