Problem 8 - Essential Calculus

Question

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(\sec ^{2} x-\cos x\right) d x $$

Step-by-Step Solution

Answer

The sketch should show two curves representative of \( \sec^{2}x \) and \( \cos{x} \) between -π/4 and π/4. The shaded region between these two curves represents the integral given.

Step 1: Graph of \( \sec^{2}x \)

Firstly, sketch the graph of \( \sec^{2}x \) which is the same as \( 1 / \cos^{2}x \) on the interval [-π/4, π/4]. Please note that \( \sec^{2}x \) is always greater than or equal to 1.

Step 2: Graph of \( \cos{x} \)

After graphing the first function, let's sketch the second function \( \cos{x} \) on the same interval [-π/4, π/4]. Keep in mind that for this interval, \( \cos{x} \) declines from \/2\/2 to \/2\/2.

Step 3: Shading the Area

Now that both functions are graphed on the same interval, shade the area between them. This shaded region represents the integral of the difference of these two functions on the given interval.