Question

Find area of region bounded by two curves?

Answer 1

Here we have got two functions f(x) and g(x) which defines two curves in the above figure. We need to find the area of the blue stripped region.

This can be achieved using the steps below:

Step 1:  Find the intersection points by equating f(x) to g(x)

Example:  

$f(x)=x^2+16x+1$  

$g(x)=-x^2+1$  

Equating f(x) and g(x), we will get a quadratic equation which is to be solved to get the intersection points

$x^2+16x+1=-x^2+1$

$2x^2+16x=0$  

x=0 and x= -8  

Step 2:  Using the points of intersection as the limits of integration, set up the definite integral and compute it to determine the area between the curves.

Area = $\mid\int\limits^0_{-8} (f(x) - g(x)) dx\mid=\mid\int\limits^0_{-8} ((x^2+16x+1)-(-x^2+1))dx\mid$

 $\mid\int\limits^0_{-8}(2x^2+16x)dx\mid=\mid(\frac{2x^3}{3}+\frac{16x^2}{2})\frac {0}{-8}\mid=\mid-\frac{512}{3}\mid=\frac{512}{3}$

Area can never be negative, that is why we are taking the absolve value of the integral