integrals for area between curves

Question

what integral formula would represent the area between the curves f(x)=-x²+2x+3 and g(x)=x+1 on the interval [-1,2]?

Answer 1

The area under f(x) in the interval [-1,2] is the integral of f(x) over the interval.

 

The area under g(x) in the interval [-1,2] is the integral of g(x) over the interval.

 

The difference between the two integrals is the area between the curves over the interval.

 

Area between the curves

 

∫[f(x)-g(x)]dx

 

∫[(-x²+2x+3)-(x+1)]dx      over [-1,2]

 

[-(x³/3)+x²+3x-(x²/2)+x]

 

[-(x³/3)+x²/2+2x]      evaluated with x=2  and then with x=-1  and subtract the latter from the former.

 

[-(8/3)+2+4]-[(1/3)+1/2-2]

 

3.5     if I did the arithmetic right.

 

  

Answer 2

Area=Integral((-x^2+2x+3)-(x+1))dx

 

Take limits from -1 to 2