Es la forma como se representa en calculo integral a la sumatoria infinita que es la siguiente notación: \( f\left(x_{1}\right) \Delta x_{1}+f\left(x_{2}\right) \Delta x_{2}+f\left(x_{3}\right) \Delta x_{3}+\ldots+f\left(x_{n-1}\right) \Delta x_{n-1}+f\left(x_{n}\right) \Delta x_{n}=\sum_{i=1}^{n} f\left(x_{i}\right) \Delta x_{i} \)

\( e^{(4+i)x} \)

\( \int_{0}^{5} f(x) d x \), where \( f(x)=\left\{\begin{array}{ll}5 & \text { if } x \leq 3 \\ 3 x-4 & \text { if } x>3\end{array}\right. \)

(a) \( \sum_{n=1}^{\infty} 2\left(\frac{4}{3}\right)^{n-1} \)

Suppose \( \int_{2}^{4} f(x) d x=-8 \int_{2}^{7} f(x) d x=6 \), and \( \int_{2}^{7} g(x) d x=5 \). Evaluate the following integrals. \( \int_{2}^{7} 6 g(x) d x=30 \) \( \int_{2}^{7}[g(x)-f(x)] d x=-1 \) (Simplify your answer) (Simplify your answer) \( \int_{2}^{7}[5 g(x)-f(x)] d x=\square \) (Simplify your answer.)

Complete the following steps for the given function interval, and value of \( n \). a. Sketch the graph of the function on the given interval. b. Calculate \( \Delta x \) and the grid points \( x_{0} x_{1} \quad x_{n} \) c. Illustrate the left and right Riemann sums, and determine which Riemann sum underestimates and which sum overestimates the area under the curve d. Calculate the left and right Riemann sums. \[ f(x)=\cos (x-5 \pi / 8) \text { on }[\pi / 2 \pi] n=4 \]