Answer:
[tex]\(A = A_1 - A_2 = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\)[/tex]
Explanation:
Step 1: Find the points of intersection between the two curves by setting the equations equal to each other and solving for t. = [tex]\(t = \pm \frac{\pi}{4}\)[/tex]
Step 2: Convert the given equations from polar to rectangular form to find the corresponding x and y values at the points of intersection. = [tex]\((x, y) = (\pm \frac{\sqrt{2}}{2}, \pm \frac{\sqrt{2}}{2})\)[/tex]
Step 3: Find the area of the region enclosed by the curve r = 2cos(t) using the formula for the area of a polar region: [tex]\(A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 dt\).[/tex] = [tex]\(A_1 = \frac{1}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (2cos(t))^2 dt = 2 \int_{0}^{\frac{\pi}{4}} cos^2(t) dt = \pi\)[/tex]
Step 4: Find the area of the region enclosed by the curve r = √(cos(2t)) using the formula for the area of a polar region: [tex]\(A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 dt\).[/tex] = [tex]\(A_2 = \frac{1}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} (\sqrt{cos(2t)})^2 dt = \frac{1}{2} \int_{0}^{\frac{\pi}{4}} cos(2t) dt = \frac{\pi}{4}\)[/tex]
Step 5: Find the exact area of the shaded portion by subtracting the area of the smaller region from the area of the larger region. = [tex]\(A = A_1 - A_2 = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\)[/tex]
