Question

consider the following limit of riemann sums of a function f on a,b. identify f and express the limit as a definite integral. the limit, expressed as a definite integral, is $lim_{delta
ightarrow0}sum_{k = 1}^{n}x_{k}^\tan x_{k}^{2}delta x_{k};2,3$

Explanation:

Step1: Recall Riemann sum definition

$\int_a^b f(x)dx = \lim_{\Delta \to 0} \sum_{k=1}^n f(x_k^*) \Delta x_k$

Step2: Identify function and interval

Given sum: $\sum x_k^* \tan^2(x_k^*) \Delta x_k$, interval [2,3]

Step3: Form definite integral

$f(x)=x\tan^2(x)$, so integral is $\int_2^3 x\tan^2(x)dx$

Answer:

$\int_{2}^{3} x \tan^2(x) \, dx$