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

Show final answer

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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