Solving the area of a 2 dimensional shape

Question

What is the area of the 2-dimensional shape given by the inequality x^2 ≤ y ≤ exp(–x^2), rounded to 4 decimal digits?

I tried to solve it but i couldn't, even google was not of a great help , so please help me thank you so much in advance

Answer 1

You can first try to solve x^2 = exp(-x^2) numerically. Say the solution is x*.

Then compute the integral of exp(-x^2) between 0 and x*, and substract the integral of x^2 on the same interval (which is x*^3/3).

Concerning the first integral, you might want to have a look at erf function implemented in math and numpy library in python for instance.

Answer 2

The shape is symmetrical about the y axis, so you can start at x = 0 and sum the rectanglar areas 2(exp(-x^2) - x^2)dx, incrementing x by dx until exp(-x^2) <= x^2.

Choose dx to be small enough to give you the accuracy you need.