Use definite integrals and Fundamental Theorem of Calculus to find the exact area of the region between the graphs of $f ( x ) = x - 2$ and $g ( x ) = x ^ { 2 } - 3 x - 2$ from x = -1 to x = 4.
A) 9
B) 13
C) 10
D) 11

Question

Quizplus · Accepted Answer

The area between the two curves from $x = -1$ to $x = 4$ can be found by integrating the difference of the functions $f(x) - g(x)$ over this interval. The difference is $(x - 2) - (x^2 - 3x - 2) = -x^2 + 4x$. The definite integral of this expression from $-1$ to $4$ gives the area:$$\int_{-1}^{4} (-x^2 + 4x) dx = \left[ -\frac{1}{3}x^3 + 2x^2 \right]_{-1}^{4} = \left( -\frac{1}{3}(4)^3 + 2(4)^2 \right) - \left( -\frac{1}{3}(-1)^3 + 2(-1)^2 \right) = -\frac{64}{3} + 32 + \frac{1}{3} + 2 = 13.$$

Use Definite Integrals and Fundamental Theorem of Calculus to Find f(x)=x−2f ( x ) = x - 2f(x)=x−2

Use Definite Integrals and Fundamental Theorem of Calculus to Find $f ( x ) = x - 2$