Question

Let R be the region in the first quadrant enclosed by the graphs of `y=e^(-x^2)`, `y=1-cosx`, and the y axis, how do you find the area of the region R?

Answer 1

`R approx .5909624501`

Explanation:

I graphed the two functions and also roughly highlighted the area we are trying to calculate:

Overview: Find the integral from zero to first intersection point of the two functions of the upper function minus the lower function

To find intersection point: set two equations equal to each other and solve using calculator ():
`e^(-x^2)=1-cosx`
(On TI 83 to TI 84+CE: either use Numerical solver under MATH menu, or graph the functions and use intersect under CALC menu)

`x approx .94194`

Area of `R`
`R = int_0^(.94194)(e^(-x^2))-(1-cosx)dx`

Plug this into graphing calculator to get:
`R approx .5909624501`