Question

How about solution. ( I = ?)

Answer 1

`I = -10^4 e^(-2) sin(2)`

Explanation:

We wish to know what the following integral evaluates to:

`I = int_0^2 int_0^1 -10^4 cos(2x) e^(-2) dx dz`

Start by removing the constants from the integrand.

`I = -10^4 e^(-2) int_0^2 int_0^1 cos(2x) dx dz`

We will let `C = -10^4 e^(-2)` so that our integral is visually easier to work with.

Integrate `cos(2x)` with respect to `x` and evaluate from 0 to 1.

`I = C int_0^2 [1/2 sin(2x)]_0^1 dz = C/2 int_0^2 sin(2) dz`

Integrate the (constant) term `sin(2)` with respect to `z` and evaluate from 0 to 2.

`I = C/2 [sin(2)z]_0^2 = C/2 2sin(2) = Csin(2)`

Finally, substitute our original constants back into `C`.

`I = -10^4 e^(-2) sin(2)`

This is our final answer.

Answer 2

Given

`I = int_0^2 int_0^1 -10^4 cos(2x) e^(-2) dx dz`
`=>I = -10^4 e^(-2) int_0^2 int_0^1 cos(2x) dx dz`

First Integrate outer integral with respect to `z`

`I = -10^4 e^(-2) | (int_0^1 cos(2x) dx)z|_0^2`
`=>I = -10^4 e^(-2)xx2 int_0^1 cos(2x) dx`

Now Integrate with respect to `x`

`I = -10^4 e^(-2)xx2 | 1/2 sin(2x) |_0^1`
`=>I = -10^4 e^(-2) sin2`