Answer to Question #101491 in Calculus for Mal

Question #101491
1. Calculate the numerical value of the following integral
∫_(-4)^2[(2-x^2 )-(3x+2)] dx

2. Calculate the curves f (x) = 2 - x2 and g (x) = 3x + 2 total area A between [-4,2]. Compare to the previous problem 1. and notice that there will be a different answer to these! Why came a different answer?
1
Expert's answer
2020-01-21T09:06:17-0500

1.

"\\int_{-4}^2[2-x^2-(3x+2)]dx=\\int_{-4}^2(-x^2-3x)dx=(-\\frac{x^3}{3}-\\frac{3x^2}{2})|_{x=-4}^{x=2}=-6."


2.

"A=\\int_{-4}^{-3}[(3x+2)-(2-x^2)]dx+\\int_{-3}^0[2-x^2-(3x+2)]dx+\\\\+\\int_0^2[(3x+2)-(2-x^2)]dx="

"=\\int_{-4}^{-3}(x^2+3x)dx+\\int_{-3}^0(-x^2-3x)dx+\\int_0^2(x^2+3x)dx="

"=(\\frac{x^3}{3}+\\frac{3x^2}{2})|_{x=-4}^{x=-3}+(-\\frac{x^3}{3}-\\frac{3x^2}{2})|_{x=-3}^{x=0}+(\\frac{x^3}{3}+\\frac{3x^2}{2})|_{x=0}^{x=2}=\\frac{11}{6}+\\frac{9}{2}+\\frac{26}{3}=15."

Answers are different because the area between two curves is always positive and in our case

includes three parts:

-4<x<-3 where g(x)>f(x)

-3<x<0 where f(x)>g(x) and

0<x<2 where g(x)>f(x).



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