Question

Suppose f(x) is a decreasing. positive function and we use different methods to approximate the integral of f over the interval [a,b]. Let Uf​(P) cenote the upper sum and Lf​(P) denote the lower sum with respect to a partiton P, of 30 subintervals with equal length LetM30​ represent the Riemann sum with midpoint approach with 30 subintervals of equal length. Which of the following statements is certaintly true?

O Uf​(P) ≤ ∫ₐᵇ​f(x)dx≤Lf​(P)
O Lf​(P) ≤ ∫ₐᵇ​f(x)dx≤Uf​(P)
O Uf​(P) < M30​O M30​ =1/2​(Lf​(P)+Uf​(P))
O None of the above

Answer 1

For a decreasing, positive function over an interval [a,b], the statement 'Lf(P) ≤ ∫ᵃᵂ f(x)dx ≤ Uf(P)' is certainly true, representing that the lower sum is less than or equal to the actual integral, which in turn is less than or equal to the upper sum.

option b is the correct

Step-by-step explanation:

The question pertains to the properties of Riemann sums and the approximation of definite integrals for a decreasing, positive function over an interval [a,b]. Given a partition P of 30 equal subintervals, we have upper sums Uf(P), lower sums Lf(P), and midpoint sums M30

For a decreasing, positive function, the following is certainly true:

• Lf(P) will be less than or equal to the actual integral of f over the interval [a,b], that is, Lf(P) ≤ ∫ᵃᵂ f(x)dx.
• The actual integral will be less than or equal to the upper sum, ∫ᵃᵂ f(x)dx ≤ Uf(P).
• Therefore, Lf(P) ≤ ∫ᵃᵂ f(x)dx ≤ Uf(P) is the statement that is certainly true.

Since f(x) is a decreasing function, the left endpoint of each subinterval in the partition will give the maximum value of f(x) for that subinterval - hence, the upper sum overestimates the integral. The right endpoint will give the minimum value, so the lower sum underestimates the integral. The midpoint is often a better approximation but there's no guarantee it will equal the average of the upper and lower sums, thus M30 = 1/2(Lf(P) + Uf(P)) is not necessarily true.