Describe how definite integrals can be used to find the total distance traveled given a velocity function.

\( \operatorname { lol } _ { x \rightarrow 0 } f x = \sqrt { 1 + 3 x } - 1 \)

38. Use integration formulas to solve each integral. You may have to use algebra, educated guess-and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. (a) (10 points) \( \int\left(x^{2}-3 x^{8}-7\right) d x= \) (b) (10 points) \( \int 3 \cos (x) d x= \) (c) (10 points) \( \int x^{2}\left(x^{3}+1\right) d x= \)

\begin{tabular}{|l|}\hline 3. \( \begin{array}{l}\text { Calcula el volumen } \\ \text { del sólido de } \\ \text { revolución que se } \\ \text { forma de girar sobre } \\ \text { el eje } x \text { las funciones } \\ f(x)=2 x^{2}+\frac{1}{2} y \\ g(x)=x \text { con límites } \\ {[0,3]}\end{array} \) \\ \hline\end{tabular}

(10 points) Use the Fundamental Theorem of Calculus to find the exact values of the definite integral. \( \int_{1}^{3} 3\left(2^{x}\right) d x= \)

4. (8 points) Calculate the following limit. Write DNE if the limit does not exist \( \lim _{x \rightarrow-1}\left(x^{2}-\frac{3 x}{x+2}\right)= \) 5. (16 points) Use the following phrases to complete the statements below to give