91Ó°ÊÓ

Without sigma notation. Then evaluate them. \(\sum_{k=1}^{2} \frac{6 k}{k+1}\)

Use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. $$f(x)=x^{2} \text { between } x=0 \text { and } x=1$$

Evaluate the integrals. $$\int_{0}^{2} x(x-3) d x.$$

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int 2(2 x+4)^{5} d x, \quad u=2 x+4$$

Express the limits as definite integrals. $$\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} c_{k}^{2} \Delta x_{k}, \text { where } P \text { is a partition of }[0,2]$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals. $$\begin{aligned} &\text { a. } \int_{0}^{3} \sqrt{y+1} d y\\\ &\text { b. } \int_{-1}^{0} \sqrt{y+1} d y \end{aligned}$$

Without sigma notation. Then evaluate them. \(\sum_{k=1}^{3} \frac{k-1}{k}\)

Use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. $$f(x)=x^{3} \text { between } x=0 \text { and } x=1$$

Evaluate the integrals. $$\int_{-1}^{1}\left(x^{2}-2 x+3\right) d x.$$

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int 7 \sqrt{7 x-1} d x, \quad u=7 x-1$$