Find the area of the polar region enclosed by f(\theta) = \sec (\theta) for -\frac{\pi}{4} <...


Find the area of the polar region enclosed by {eq}f(\theta) = \sec (\theta) {/eq} for {eq}-\frac{\pi}{4} < \theta < \frac{\pi}{4}. {/eq}

Calculating the Area Bounded by a Polar Curve


Consider a polar curve given by the equation :

{eq}r = f(\theta) \\ \text{Area bounded by the curve between } \theta = \theta_1 \text{ and } \theta = \theta_2 \text{ is given by :} \\ {/eq}

$$A = \int_{\theta_1}^{\theta_2} r^2 \ d\theta \\ \Rightarrow A = \int_{\theta_1}^{\theta_2} f^2(\theta) \ d\theta \\ $$

The procedure of solving this definite integral depends on the nature of polar function. We will make use of this formula to find the bounded area in the given problem.


Answer and Explanation: 1

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Given polar curve :

{eq}r = f(\theta) = \sec{\theta} \\ {/eq}

Therefore, the area enclosed within the given region is given by :


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Learn more about this topic:

Evaluating Definite Integrals Using the Fundamental Theorem


Chapter 16 / Lesson 2

The fundamental theorem of calculus makes finding your definite integral almost a piece of cake. See how the definite integral becomes a subtraction problem after applying the fundamental theorem of calculus.

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