estimate the value of the integral

Question

Given that 3≤f(x)≤5 for -3≤x≤8, estimate the value of ∫_-3⁸f(x)dx

___≤∫_-3⁸f(x)dx≤___

Answer 1

The value of a definite integral (integral with boundaries, like here) is equal to the area under the curve f(x) on the specified interval.

The smallest value the function can have is 3, it can equal to 3 on the whole interval from -3 to 8, thus the area of the rectangle between the f(x)=3 and the x-axis is 3x11 = 33 (11 is the length from -3 to 8).

The largest value the function can have is 5. If f(x)=5 for the whole interval from -3 to 8, the area of the rectangle is 5x11=55.

Thus, the lower boundary is 33 and the upper one is 55.