91Ó°ÊÓ

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]

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). a. \(\int_{0}^{\pi / 4} \tan x \sec ^{2} x d x\) b. \(\int_{-\pi / 4}^{0} \tan x \sec ^{2} x d x\)

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{4} \cos k \pi$$

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. \(f(x)=1 / x\) between \(x=1\) and \(x=5.\)

Evaluate the integrals. $$\int_{-2}^{2} \frac{3}{(x+3)^{4}} d x$$

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

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{5} \sin k \pi$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). a. \(\int_{0}^{\pi} 3 \cos ^{2} x \sin x d x\) b. \(\int_{2 \pi}^{3 \pi} 3 \cos ^{2} x \sin x d x\)

Interpreting Limits of Sums as Integrals Express the limits in Exercises \(1-8\) as definite integrals. \(\lim _{|P| \rightarrow 0} \sum_{i=1}^{n}\left(\frac{1}{c_{i}}\right) \Delta x_{i},\) where \(P\) is a partition of [1,4]

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. \(f(x)=4-x^{2}\) between \(x=-2\) and \(x=2.\)