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 σ (x_k*)^4 δx_k, 9,13
δ→0 k = 1

Explanation:

Step1: Identify function from sum

The Riemann sum has \( (x_k^*)^4 \), so \( f(x) = x^4 \).

Step2: Determine interval [a,b]

Given interval is [9,13], so \( a=9 \), \( b=13 \).

Step3: Express as definite integral

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

Answer:

\(\int_{9}^{13} x^4 \, dx\)