Question

Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

Accepted Answer

First, identify the curves that bound the region: the upper curve is given by $$y = 2 \cos x$$ and the lower curve by $$y = \sec x$$ over the interval $$-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}. $$Next, set up the integral for the area between the two curves. The area $$A is $$given by the integral of the difference between the upper and lower functions over the given interval:

$$A = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \left( 2 \cos x - \sec x ight) \, dx. $$Before integrating, verify that $$2 \cos x is $$indeed greater than $$\sec x on $$the interval $$[-\frac{\pi}{4}, \frac{\pi}{4}] to $$ensure the correct order of subtraction for the area calculation. Proceed to integrate each term separately:

$$\int 2 \cos x \, dx$$ and $$\int \sec x \, dx. $$Recall that the integral of $$\cos x is$$ $$\sin x$$, and the integral of $$\sec x is$$ $$\ln |\sec x + 	an x|. $$Finally, evaluate the definite integral by substituting the limits $$x = \frac{\pi}{4}$$ and $$x = -\frac{\pi}{4}$$ into the antiderivative expression, and subtract the results to find the area.

Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.

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Key Concepts

Definite Integrals for Area Calculation

Understanding and Comparing Trigonometric Functions

Interval and Domain Considerations