Sketch the region enclosed by the given curves.y = 4 cos(7x),    y = 4 − 4 cos(7x),    0 ≤ x ≤ 𝜋/7 Find its area.

To find the area enclosed by the curves, we first need to understand that the area between two curves is given by the integral of the absolute difference of the two functions over the given interval.

The given functions are y = 4cos(7x) and y = 4 - 4cos(7x). The interval is from 0 to 𝜋/7.

First, let's find the difference between the two functions:

y2 - y1 = (4 - 4cos(7x)) - 4cos(7x) = 4 - 8cos(7x)

Since we are looking for the absolute difference, we can ignore the negative sign if it appears.

Next, we integrate this difference over the given interval:

∫ from 0 to 𝜋/7 of |4 - 8cos(7x)| dx

This integral can be split into two parts:

∫ from 0 to 𝜋/7 of 4 dx - ∫ from 0 to 𝜋/7 of 8cos(7x) dx

The first integral evaluates to 4x evaluated from 0 to 𝜋/7, which is 4𝜋/7.

The second integral evaluates to (8/7)sin(7x) evaluated from 0 to 𝜋/7. Since sin(0) = 0 and sin(𝜋) = 0, this integral evaluates to 0.

Therefore, the area enclosed by the curves is 4𝜋/7 square units.