## Upper and Lower Estimates for Integrals

Some integrals cannot be calculated exactly. A good example is The integral exists and is equal to the area A below. In examples like this it is useful to be able to calculate lower and uppoer bounds for the value of an integral. We can find a lower bound by dividing the interval of integration on the axis into sunbintervals and finding the minimum value of the function, on this interval. Form rectangles with base of length and height and area and add these to get a lower bound for the value of the integral, To find an upper bound find the maximum value of the function, F-i on each subinterval Form rectangles of base and height F-i and area and add these to get a upper bound for the value of the integral, To illstrate, find upper and lower bounds for on by dividing the interval into 5 subintervals A lower bound is found from the diagram below:  An upper bound for the integral is found from the diagram below.  Compare these bound with the actual value  