Question

How do you find the area between `y=e^x` and `y=e^-x` and x=1?

Answer 1

Start by finding the intersection point of the two functions.

`e^x = e^-x`

`e^x - e^-x = 0`

`e^x - 1/e^x= 0`

`e^(2x) - 1 = 0`

`e^(2x)= 1`

`2xlne = ln1`

`2x= 0`

`x = 0`

We also know through end behaviour of the function that `y= e^x` will be above `y = e^-x`. So, we determine the area of `y= e^x` in the interval `0 ≤ x ≤1` and then subtract the area of `y= e^-x` in the interval `0 ≤ x ≤1`.

`int_0^1(e^x - e^-x)dx = e^x + e^-x|_0^1 = e^1 +1/e - (1+ 1) = -2 + 1/e + e`

This can be approximated to `1.086" u"^2`.

Hopefully this helps!