Question

Question #dc81f

Answer 1

`5(1/(2e^2)+7)` or `~35.33834`

Explanation:

let's make the exponential decay graph `f(x)` graph{e^(-0.4x) [-7.024, 7.024, -3.51, 3.514]}
and the linear graph `g(x)`
graph{2.4x+1 [-7.024, 7.024, -3.51, 3.514]}

Judging by the graphs, the linear graph will be the "upper curve" from `0` to `5` since it is greater than the exponential decay graph at all points `(0,5]`

So it will be integration of upper area minus lower area for the domain, or `int_0^5g(x)-f(x)dx`.
Then you substitute the equations, so
`int_0^5(2.4x+1)-(e^(-0.4x))dx`
=`int_0^5 2.4x+1-e^(-0.4x)dx`

Then you can integrate and solve using the 2nd Fundamental Theorem of Calculus.

`int_0^5 2.4x+1-e^(-0.4x)dx`
=`(1.2x^2+x+2.5e^(-0.4x))|_0^5`
=`1.2*5^2+5+2.5e^(-0.4*5)-(0)`
=`5(1/(2e^2)+7)` or `~35.33834`

Ask if you need any clarification on this. Thanks for asking though.