91Ó°ÊÓ

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

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

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. \(f(x)=x^{2}\) between \(x=0\) and \(x=1.\)

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

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). a. \(\int_{0}^{3} \sqrt{y+1} d y\) b. \(\int_{-1}^{0} \sqrt{y+1} d y\)

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{|P| \rightarrow 0} \sum_{k=1}^{n} c_{k}^{2} \Delta x_{k},\) where \(P\) is a partition of [0,2]

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

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

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{|P| \rightarrow 0} \sum_{k=1}^{n} 2 c_{k}^{3} \Delta x_{k},\) where \(P\) is a partition of [-1,0]

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