Question

let (f) be the function given by (f(x)=\frac{x^{3}+1}{sqrt{x^{2}+x}}). it is known that (f) is increasing on the interval (2,8). let (r_{3}) be the value of the right riemann sum approximation for (int_{2}^{8}f(x)dx) using 3 intervals of equal length. which of the following statements is true?
(a) (r_{3}=103.385) is an underestimate for (int_{2}^{8}f(x)dx).
(b) (r_{3}=103.385) is an overestimate for (int_{2}^{8}f(x)dx).
(c) (r_{3}=216.952) is an underestimate for (int_{2}^{8}f(x)dx).
(d) (r_{3}=216.952) is an overestimate for (int_{2}^{8}f(x)dx).

Explanation:

Step1: Recall right - Riemann sum property

For an increasing function on an interval $[a,b]$, the right - Riemann sum overestimates the definite integral $\int_{a}^{b}f(x)dx$.

Step2: Analyze the given function and sum

The function $f(x)=\frac{x^{3}+1}{\sqrt{x^{2}+x}}$ is increasing on $[2,8]$, and $R_{3}$ is a right - Riemann sum. So $R_{3}$ overestimates $\int_{2}^{8}f(x)dx$.

Answer:

B. $R_{3}=103.385$ is an overestimate for $\int_{2}^{8}f(x)dx$.