Final answer:
A general-purpose procedure for estimating integrals includes partitioning the interval, calculating upper and lower sums for those partitions, and then averaging them once the sums are within a desired tolerance of 2ε.
Step-by-step explanation:
The procedure to estimate integrals of decreasing functions using the method of upper and lower sums with a uniform partition is as follows:Determine the width of each partition by dividing the interval [a, b] by n.For each subinterval, calculate the upper sum by taking the function value at the left endpoint (for decreasing functions) and multiply by the partition widthCalculate the lower sum by taking the function value at the right endpoint and multiply by the partition widthContinue increasing n until the difference between sum upper and sum lower is less than 2εOnce the condition is met, return the average of sum upper and sum lower as an estimate of the integral.
To test this procedure, one would apply it to the sine integral with ε = 0.000005 to ensure the accuracy of the estimate within the specified tolerance.To estimate integrals of decreasing functions by the method of upper and lower sums with a uniform partition, we can use the procedure Integral(f, a, b, ε, n, sum lower, sum upper). This procedure takes in the function name f, the endpoints of the interval a and b, the tolerance ε, and the lower and upper sums. It determines the value of n such that sum upper - sum lower < 2ε. The procedure then returns the average of the upper and lower sums.For example, let's test this procedure on the sine integral using ε = 0.5 × 10-5. The calling sequence would be Integral(sin, a, b, 0.5 × 10-5, n, sum lower, sum upper). This procedure will provide an estimation of the integral of the sine function within the given tolerance.