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Definite Integral – Finding area between a polynomial and a line – Exercise 7006

Exercise

Find the area of the region bounded by the graphs of the equations:

y=x^2, y=2x+3

Final Answer


S=10\frac{2}{3}

Solution

First, we find out how the area looks like:

x^2=2x+3

x^2-2x-3=0

(x+1)(x-3)=0

x=-1, x=3

The area looks like this:

S=\int_{-1}^3 2x+3-x^2 dx=

=[2\cdot\frac{x^2}{2}+3x-\frac{x^3}{3}]_{-1}^3=

=[x^2+3x-\frac{x^3}{3}]_{-1}^3=

=3^2+3\cdot 3-\frac{3^3}{3}-({(-1)}^2+3\cdot (-1)-\frac{{(-1)}^3}{3})=

=9+9-9-(1-3+\frac{1}{3})=

=9-1+3-\frac{1}{3}=

=10\frac{2}{3}

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