To determine

To find:

The interval a, b for which the given integral is a maximum

Answer

-1, 2

Explanation

1) Given:

∫ab2+x-x2dx

2) Calculation:

Graph of 2+x-x2

So, the graph of the given integral is a downward parabola. The integral is the net area between the curve and the x-axis.

The maximum value of the integral is over the interval between the zeros of the parabola because only there the area is positive.

So, find out the roots of the parabola.

0=2+x-x2

x2-x-2=0

x-2x+1=0

x=2  or x=-1

The interval is -1, 2.

Conclusion:

Therefore, the value of ∫ab2+x-x2dx is maximum in the interval -1, 2.