Sketch the region enclosed by the given curves.y = 4 cos(𝜋x),    y = 8x2 − 2 Find its area.

To find the area enclosed by the curves y = 4cos(πx) and y = 8x² - 2, we first need to find the points of intersection.

Step 1: Set the two equations equal to each other and solve for x.

4cos(πx) = 8x² - 2

This equation is transcendental and may be difficult to solve analytically. However, we can solve it graphically or numerically.

Step 2: Sketch the region enclosed by the curves.

The function y = 4cos(πx) is a cosine function with amplitude 4 and period 2. The function y = 8x² - 2 is a parabola opening upwards with vertex at (0, -2).

Step 3: Find the area enclosed by the curves.

The area A between the curves from x = a to x = b is given by the integral from a to b of the absolute difference of the functions.

A = ∫ from a to b |4cos(πx) - (8x² - 2)| dx

This integral may be difficult to solve analytically, but it can be approximated numerically.

Remember to calculate the integral for each bounded region separately if there are more than one.