Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.
A) -158
B) 12
C) 80
D) 82
E) -160

Question

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. 
A) -158 
B) 12 
C) 80 
D) 82 
E) -160

Quizplus · Accepted Answer

Since the integral represents an area, we can sketch the area bounded by the given curve and the x-axis. From the interval −2 to 3, the curve is the line y = 2x + 4, and from 3 to 4, the curve is the parabola y = −(x − 4)^2 + 12. The intersection point of these curves is (3, 10). Therefore, the region is a trapezoid with height 6 (from x = −2 to x = 4) and bases of length 8 and 16 (the height of the trapezoid at x = −2 is 0, so its base is 2(4) = 8, while the height of the trapezoid at x = 4 is again 0, so its base is 2(8) = 16). Using the formula for the area of a trapezoid, we obtain A = (1/2)(8 + 16)(6) = 72. The integral is then equal to the area of this region, which is 80. Therefore, the answer is C.

Sketch the Region Whose Area Is Given by the Definite