Problem

Use a limit of Riemann sums to evaluate the definite integral. \[ \begin{aligned} \int_{3}^{5}(4 x-1) d x & =\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \\ & =\lim _{n \rightarrow \infty} \\ & = \end{aligned} \] Hint: This list of summation rules may be helpful.

Solution

Step 1 :Set up the Riemann sum for the function \( f(x) = 4x - 1 \) over the interval \([3, 5]\)

Step 2 :Use the limit definition of the definite integral, taking the limit as \( n \) approaches infinity of the sum of \( f(x_i) \times \Delta x \)

Step 3 :Use the right endpoint of each subinterval as the sample point

Step 4 :Calculate the definite integral using the limit of Riemann sums

Step 5 :The final answer is \(\boxed{30}\)

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