Question

How to find area bounded by a curve and a line?

The points of intersection of the line and the curve are ( -1 , 9 ) and ( 3 , 1 )

Answer 1

The reference Area Between Curves gives the equation:

`A = int_a^bf(x)-g(x)dx`

where `f(x)>=g(x) and a<=x <=b`

Explanation:

In your case, the diagram shows that f(x) is the line but we must write is as y in terms of x:

`y = 7-2x`

`f(x) = 7 - 2x`

The quadratic is g(x):

`y = (x-2)^2`

`g(x) = (x-2)^2`

Expand the square:

`g(x) = x^2-2x+4`

You have provided the limits of integration by giving the x coordinates where the functions intersect:

`a = -1` and `b=3`

Start with the equation:

`A = int_a^bf(x)-g(x)dx`

Substitute the values of `f(x),g(x),a, and b` into the equation:

`A = int_-1^3 7 -2x -(x^2-2x+4)dx`

Before we integate, I shall simplify the integrand:

`A = int_-1^3 7 -2x -x^2+2x-4dx`

`A = int_-1^3 -x^2+3dx`

Integrate

`A = [-1/3x^3+3x]_-1^3`

Evaluate at the limits:

`A = [-1/3(3)^3+3(3)] - [-1/3(-1)^3+3(-1)]`

`A = 8/3`