Question

Find the area?

Find the area bounded by the graph of y=cosh2x and the x-axis from x=-1 to to x=2.

Answer 1

`A=(sinh4+sinh2)/2`

Explanation:

Let `A` be the required area.

`A = int_(x=-1)^(x=2) cosh(2x)"d"x`

Remember that

`coshz=(e^z+e^(-z))/2`

`:. A = int_(-1)^2 (e^(2x)+e^(-2x))/2"d"x`

`:. A = 1/2(int_(-1)^2 e^(2x) "d"x + int_(-1)^2 e^(-2x)"d"x)`

We will solve these by substitution: let `u_1=2x` and `u_2=-2x`.

`"d"x=("d"u_1)/2=-("d"u_2)/2`

`:. A = 1/2 (int_(-2)^4 e^(u_1) ("d"u_1)/2+int_(2)^(-4) e^(u_2)(-"d"u_2)/2)`

`:. A = 1/4 (int_(-2)^4 e^(u_1) "d"u_1-int_2^(-4) e^(u_2)"d"u_2)`

`:. A = 1/4 (e^4-e^(-2)-e^(-4)+e^(2))`

`:. A=1/4(e^4-e^(-4))+1/4(e^2-e^(-2))`

`:. A = (sinh4+sinh2)/2`