Find the area of the region enclosed by the graphs of and .

Question

Find the area of the region enclosed by the graphs of  and  . ​

Quizplus · Accepted Answer

To find the region enclosed by the graphs, we need to find the x-coordinates of the points of intersection of the two curves. Setting them equal and solving, we get:

11ea984b_b091_418f_9029_ef04926eed1d_TB4005_11 = 11ea984b_b091_68a0_9029_2b5f109db57d_TB4005_11

x^2 + 5x + 6 = x^2 - x - 6

6x + 12 = 0

x = -2

Therefore, the two curves intersect at x=-2.

We can now integrate the difference between the curves over the interval [-2,3] to find the enclosed area:

∫[-2,3] (11ea984b_b091_418f_9029_ef04926eed1d_TB4005_11 - 11ea984b_b091_68a0_9029_2b5f109db57d_TB4005_11) dx

= ∫[-2,3] (6x + 8) dx

= [3x^2 + 8x] from x=-2 to x=3

= 31

Therefore, the enclosed area is 31, which is answer choice C.

Find the Area of the Region Enclosed by the Graphs