Sketch the region enclosed by the curves $y = x + 2 , y = 9 - x ^ { 2 } , x = - 2$, and $x = 2$. Find the area of the region correct to two decimal places.
Select the correct answer.
A) $\quad 22.67$
B) $32.67$
C) $20.67$
D) $52.67$
E) $62.67$
F) $42.67$

Question

Quizplus · Accepted Answer

The region is bounded by the curves $y = x + 2 , y = 9 - x ^ { 2 } , x = - 2$, and $x = 2$. The area of the region is given by the integral$$\int_{-2}^{2} (9 - x^2) - (x + 2) dx = \int_{-2}^{2} (7 - x^2 - x) dx = \left[ 7x - \frac{x^3}{3} - \frac{x^2}{2} \right]_{-2}^{2} = 22.66666667$$Therefore, the area of the region is approximately 22.67 square units.

Sketch the Region Enclosed by the Curves y=x+2,y=9−x2,x=−2y = x + 2 , y = 9 - x ^ { 2 } , x = - 2y=x+2,y=9−x2,x=−2

Sketch the Region Enclosed by the Curves $y = x + 2 , y = 9 - x ^ { 2 } , x = - 2$