Question

Suppose f is a function with$f\left(x\right)>0,f\text{'}\left(x\right)\le 0,\mathrm{and}f\text{'}\text{'}\left(x\right)\le 0$

(a) Put the right-sum, left-sum, trapezoid-sum, and midpoint-sum approximations of ${\int }_a^{b}f\left(x\right)dx$ in order from smallest to largest. (The order will not depend

(b) The true value of ${\int }_a^{b}f\left(x\right)dx$must lie between two of the approximations that you listed in part (a) Which two, and why?

Short answer

(a) Left (n) < Mid (n) < Trap (n) < Right (n)

(b)${\int }_a^{b}f\left(x\right)dx$must lie between trapezoid sum and midpoint sum.

Step-by-step solution

Step 1. Given information

$f\left(x\right)>0,f\text{'}\left(x\right)\le 0,f\text{'}\text{'}\left(x\right)\le 0$

Part(a) Step 2. Explanation

The left-sum approach clearly covers the most area, whereas the right-sum method covers the smallest.

Left (n) < Mid (n) < Trap (n) < Right (n) is the exact order of the methods in decreasing order of approximations.

Part(b) Step 2. Explanation

The given graph represents a concave-up function that is also monotonically lowering. As a result, the real value of the area under the graph should be somewhere between left and right sums, as well as midway and trapezoid sums.

The midpoint graph and trapezoid graphs, as seen above, are more accurately close to the real value of integrals.

As a result, the real value of ${\int }_a^{b}f\left(x\right)dx$must be somewhere between the trapezoid and midpoint sum.