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

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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

{eq}{/eq}

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.

{eq}{/eq}

Answer and Explanation: 1

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{eq}{/eq}

Given polar curve :

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

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

{eq}\d...

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Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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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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