Question

Find the area of the region bounded by the y-axis, the line y=e, and the graph of the function y=e^3x?

Answer 1

Let's start by finding the intersection point between `y = e` and `y =e^(3x)`.

`e = e^(3x)`

`1 = 3x`

`x= 1/3`

We know now that our bounds of integration will be `0` to `1/3` (since we're given that the other boundary is the y-axis, where `x = 0`).

Also, we can see that for `0 < x < 1/3`, `e > e^(3x)`. Thus our expression for area will be

`A = int_0^(1/3) e - e^(3x) dx`

`A = [ex - 1/3e^(3x)]_0^(1/3)`

`A = 1/3e - 1/3e^1 - (e(0) - 1/3e^(3(0)))`

`A = 1/3e - 1/3e+ 1/3`

`A = 1/3` square units

Hopefully this helps!