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Definite Integral – Finding area between a polynomial and a line – Exercise 7002

Exercise

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

y=-x^2+5x+6, y=-x+6

Final Answer


S=36

Solution

First, we find out how the area looks like:

-x^2+5x+6=-x+6

x^2-6x=0

x(x-6)=0

x=0, x=6

The area looks like this:

S=\int_0^6 -x^2+5x+6-(-x+6) dx=

=\int_0^6 -x^2+6x dx=

=[-\frac{x^3}{3}+6\cdot\frac{x^2}{2}]_0^6=

=[-\frac{x^3}{3}+3x^2]_0^6=

=-\frac{6^3}{3}+3\cdot 6^2-(-\frac{0^3}{3}+3\cdot 0^2)=

=-\frac{216}{3}+3\cdot 36-0=

=36

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