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

Question

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

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Answer 1 · 2017-08-15T21:49:11

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#

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#