Exercise
Find the area of the region bounded by the graphs of the equations:
y=-x^2+5x+6, y=-x+6
Final Answer
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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