Application of Integrals

Let the function, f(x)= cases-3ax2-2, & x<1\ a2+bx, & x 1 cases be differentiable for all x , where a>1, b . If the area of the region enclosed by y=f(x) and the line y=-20 is + 3, , Z, then the value of + is .

Answer

34

Solution

Differentiable at x=1: -3a-2=a2+b and -6a=b a=2, b=-12.

So f=-6x2-2 for x<1 and f=4-12x for x 1; it meets y=-20 at x=- 3 and x=2.

[figure]

Area = - 31(18-6x2) dx+ 12(24-12x) dx

=(16+12 3)+6=22+12 3

+ =34

Answer: 34