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

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

1 Answer
Aug 15, 2017

The reference Area Between Curves gives the equation:

A=baf(x)g(x)dx

where f(x)g(x)andaxb

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=72x

f(x)=72x

The quadratic is g(x):

y=(x2)2

g(x)=(x2)2

Expand the square:

g(x)=x22x+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=baf(x)g(x)dx

Substitute the values of f(x),g(x),a,andb into the equation:

A=3172x(x22x+4)dx

Before we integate, I shall simplify the integrand:

A=3172xx2+2x4dx

A=31x2+3dx

Integrate

A=[13x3+3x]31

Evaluate at the limits:

A=[13(3)3+3(3)][13(1)3+3(1)]

A=83