find the area of the region bounded by the graph of equations?

Question

y = tanh 2x, y = 0, x = 2

Answer 1

Since y = tanh(2x) and y = 0 intersect at the origin, and x = 2 is a vertical line on the right side, you're left with a triangular sort of shape, with one side that bulges.

To find the area, you have to integrate the functions, with respect to x, from 0 to 2. The shape doesn't extend beyond the interval [0, 2].

y = 0 integrates to zero.

y = tanh(2x) integrates to
1/2*ln(cosh(2x))
and when this is evaluated from 0 to 2, you get
1\2*ln(cosh(4)) - 0
= 1.65359