Chapter 5

Evaluate the sums. $$\begin{array}{lll} \text { a. } \sum_{k=1}^{10} k & \text { b. } \sum_{k=1}^{10} k^{2} & \text { c. } \sum_{k=1}^{10} k^{3} \end{array}$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{0}^{\pi / 4}(1-\sin 2 t)^{3 / 2} \cos 2 t d t$$

Graph the integrands and use known area formulas to evaluate the integrals. $$\int_{-1}^{1}(1-|x|) d x$$

Evaluate the integrals. $$\int 3 y \sqrt{7-3 y^{2}} d y$$

A power plant generates electricity by burning oil. Pollutants produced as a result of the burning process are removed by scrubbers in the smokestacks. Over time, the scrubbers become less efficient and eventually they must be replaced when the amount of pollution released exceeds government standards. Measurements are taken at the end of each month determining the rate at which pollutants are released into the atmosphere, recorded as follows. $$\begin{array}{lcccccc} \hline \text { Month } & \text { Jan } & \text { Feb } & \text { Mar } & \text { Apr } & \text { May } & \text { Jun } \\ \hline \text { Pollutant } & & & & & \\ \text { release rate } & 0.20 & 0.25 & 0.27 & 0.34 & 0.45 & 0.52 \\ \text { (tons/day) } & & & & & & \\ \hline \end{array}$$ $$\begin{array}{lcccccc} \hline \text { Month } & \text { Jul } & \text { Aug } & \text { Sep } & \text { Oct } & \text { Nov } & \text { Dec } \\ \hline \text { Pollutant } & & & & & & \\ \text { release rate } & 0.63 & 0.70 & 0.81 & 0.85 & 0.89 & 0.95 \\ \text { (tons/day) } & & & & & & \\ \hline \end{array}$$ a. Assuming a 30 -day month and that new scrubbers allow only 0.05 ton/day to be released, give an upper estimate of the total tonnage of pollutants released by the end of June. What is a lower estimate? b. In the best case, approximately when will a total of 125 tons of pollutants have been released into the atmosphere?

Evaluate the integrals. $$\int_{-\sqrt{3}}^{\sqrt{3}}(t+1)\left(t^{2}+4\right) d t$$

Evaluate the sums. a. \(\sum_{k=1}^{13} k\) b. \(\sum_{k=1}^{13} k^{2}\) c. \(\sum_{k=1}^{13} k^{3}\)

Graph the integrands and use known area formulas to evaluate the integrals. $$\int_{-1}^{1}(2-|x|) d x$$

Evaluate the integrals. $$\int \frac{1}{\sqrt{x}(1+\sqrt{x})^{2}} d x$$

Inscribe a regular \(n\) -sided polygon inside a circle of radius 1 and compute the area of the polygon for the following values of \(n:\) a. 4 (square) b. 8 (octagon) c. 16 d. Compare the areas in parts (a), (b), and (c) with the area of the circle.

Evaluate the integrals. $$\int_{\sqrt{2}}^{1}\left(\frac{u^{7}}{2}-\frac{1}{u^{5}}\right) d u$$

Evaluate the sums. $$\sum_{k=1}^{7}(-2 k)$$

Graph the integrands and use known area formulas to evaluate the integrals. $$\int_{-1}^{1}(1+\sqrt{1-x^{2}}) d x$$

Evaluate the integrals. $$\int \sqrt{\sin x} \cos ^{3} x d x$$

Evaluate the integrals. $$\int_{-3}^{-1} \frac{y^{5}-2 y}{y^{3}} d y$$

Evaluate the sums. $$\sum_{k=1}^{5} \frac{\pi k}{15}$$

Use known area formulas to evaluate the integrals in Exercises \(23-28\) $$\int_{0}^{b} \frac{x}{2} d x, \quad b>0$$

Evaluate the integrals. $$\int \sec ^{2}(3 x+2) d x$$

Use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into \(n=100,200,\) and 1000 sub intervals of equal length and evaluate the function at the midpoint of each sub interval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=\sin x \quad \text { on } \quad[0, \pi]$$

Evaluate the integrals. $$\int_{1}^{\sqrt{2}} \frac{s^{2}+\sqrt{s}}{s^{2}} d s$$

Evaluate the sums. $$\sum_{k=1}^{6}\left(3-k^{2}\right)$$

Use known area formulas to evaluate the integrals in Exercises \(23-28\) $$\int_{0}^{b} 4 x d x, \quad b>0$$

Use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into \(n=100,200,\) and 1000 sub intervals of equal length and evaluate the function at the midpoint of each sub interval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=\sin ^{2} x \quad \text { on } \quad[0, \pi]$$

Evaluate the integrals. $$\int \tan ^{2} x \sec ^{2} x d x$$

Evaluate the integrals. $$\int_{1}^{8} \frac{\left(x^{1 / 3}+1\right)\left(2-x^{2 / 3}\right)}{x^{1 / 3}} d x$$

Evaluate the sums. $$\sum_{k=1}^{6}\left(k^{2}-5\right)$$

Use known area formulas to evaluate the integrals in Exercises \(23-28\) $$\int_{a}^{b} 2 s d s, \quad 0

Use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into \(n=100,200,\) and 1000 sub intervals of equal length and evaluate the function at the midpoint of each sub interval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=x \sin \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right]$$

Evaluate the integrals. $$\int \sin ^{5} \frac{x}{3} \cos \frac{x}{3} d x$$

Evaluate the integrals. $$\int_{\pi / 2}^{\pi} \frac{\sin 2 x}{2 \sin x} d x$$

Evaluate the sums. $$\sum_{k=1}^{5} k(3 k+5)$$

Use known area formulas to evaluate the integrals in Exercises \(23-28\) $$\int_{a}^{b} 3 t d t, \quad 0

Use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into \(n=100,200,\) and 1000 sub intervals of equal length and evaluate the function at the midpoint of each sub interval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=x \sin ^{2} \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right]$$

Evaluate the integrals. $$\int \tan ^{7} \frac{x}{2} \sec ^{2} \frac{x}{2} d x$$

Evaluate the integrals. $$\int_{0}^{\pi / 3}(\cos x+\sec x)^{2} d x$$

Evaluate the sums. $$\sum_{k=1}^{7} k(2 k+1)$$

Evaluate the integrals. $$\int r^{2}\left(\frac{r^{3}}{18}-1\right)^{5} d r$$

Evaluate the integrals. $$\int_{-4}^{4}|x| d x$$

Evaluate the sums. $$\sum_{k=1}^{5} \frac{k^{3}}{225}+\left(\sum_{k=1}^{5} k\right)^{3}$$

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-46\). $$\int_{0}^{\pi / 3} \frac{4 \sin \theta}{1-4 \cos \theta} d \theta$$

Evaluate the integrals. $$\int r^{4}\left(7-\frac{r^{5}}{10}\right)^{3} d r$$

Evaluate the integrals. $$\int_{0}^{\pi} \frac{1}{2}(\cos x+|\cos x|) d x$$

Evaluate the sums. $$\left(\sum_{k=1}^{7} k\right)^{2}-\sum_{k=1}^{7} \frac{k^{3}}{4}$$

Evaluate the integrals. $$\int x^{1 / 2} \sin \left(x^{3 / 2}+1\right) d x$$

Evaluate the integrals. $$\int_{0}^{\ln 2} e^{3 x} d x$$

Evaluate the sums. $$\begin{array}{lll}\text { a. } \sum_{k=1}^{7} 3 & \text { b. } \sum_{k=1}^{500} 7 & \text { c. } \sum_{k=3}^{264} 10 \end{array}$$

Evaluate the integrals. $$\int \csc \left(\frac{v-\pi}{2}\right) \cot \left(\frac{v-\pi}{2}\right) d v$$

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

Evaluate the sums. a. \(\sum_{k=9}^{36} k\) b. \(\sum_{k=3}^{17} k^{2}\) c. \(\sum_{k=18}^{71} k(k-1)\)

Evaluate the integrals. $$\int \frac{\sin (2 t+1)}{\cos ^{2}(2 t+1)} d t$$