Chapter 5

Suppose that \(\int_{1}^{2} f(x) d x=5 .\) Find $$ \begin{array}{ll}{\text { a. } \int_{1}^{2} f(u) d u} & {\text { b. } \int_{1}^{2} \sqrt{3} f(z) d z} \\ {\text { c. } \int_{2}^{1} f(t) d t} & {\text { d. } \int_{1}^{2}[-f(x)] d x}\end{array} $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \begin{array}{l}{\int \frac{d x}{\sqrt{5 x+8}}} \\ {\text { a. Using } u=5 x+8 \quad \text { b. Using } u=\sqrt{5 x+8}}\end{array} $$

Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ 1+4+9+16 $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{\pi / 6}^{5 \pi / 6} \csc ^{2} x d x $$

Suppose that \(\int_{-3}^{0} g(t) d t=\sqrt{2} .\) Find $$ \begin{array}{ll}{\text { a. } \int_{0}^{-3} g(t) d t} & {\text { b. } \int_{-3}^{0} g(u) d u} \\ {\text { c. } \int_{-3}^{0}[-g(x)] d x} & {\text { d. } \int_{-3}^{0} \frac{g(r)}{\sqrt{2}} d r}\end{array} $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \sqrt{3-2 s} d s $$

Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{\pi / 4}^{3 \pi / 4} \csc \theta \cot \theta d \theta $$

Suppose that \(f\) is integrable and that \(\int_{0}^{3} f(z) d z=3\) and \(\int_{0}^{4} f(z) d z=7 .\) Find $$ (a)\int_{3}^{4} f(z) d z \quad \text { b. } \int_{4}^{3} f(t) d t $$

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \text { a. }\int_{-\pi / 2}^{0} \frac{\sin w}{(3+2 \cos w)^{2}} d w \quad \text { b. } \int_{0}^{\pi / 2} \frac{\sin w}{(3+2 \cos w)^{2}} d w $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int(2 x+1)^{3} d x $$

Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ 2+4+6+8+10 $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{\pi / 3} 4 \sec u \tan u d u $$

Suppose that \(h\) is integrable and that \(\int_{-1}^{1} h(r) d r=0\) and \(\int_{-1}^{3} h(r) d r=6 .\) Find $$ (a)\int_{1}^{3} h(r) d r \quad \text { b. }-\int_{3}^{1} h(u) d u $$

Distance traveled by a projectile An object is shot straight up- ward from sea level with an initial velocity of \(400 \mathrm{ft} / \mathrm{sec} .\) a. Assuming that gravity is the only force acting on the object, give an upper estimate for its velocity after 5 sec have elapsed. Use \(g=32 \mathrm{ft} / \mathrm{sec}^{2}\) for the gravitational acceleration. b. Find a lower estimate for the height attained after 5 sec.

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{0}^{1} \sqrt{t^{5}+2 t}\left(5 t^{4}+2\right) d t $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{1}{\sqrt{5 s+4}} d s $$

Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{\pi / 2}^{0} \frac{1+\cos 2 t}{2} d t $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-2}^{4}\left(\frac{x}{2}+3\right) d x $$

In Exercises \(15-18\) , use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of equal length and evaluating \(f\) at the subinterval midpoints. $$ f(x)=x^{3} \quad \text { on } \quad[0,2] $$

Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ -\frac{1}{5}+\frac{2}{5}-\frac{3}{5}+\frac{4}{5}-\frac{5}{5} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-\pi / 3}^{\pi / 3} \frac{1-\cos 2 t}{2} d t $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{1 / 2}^{3 / 2}(-2 x+4) d x $$

In Exercises \(15-18\) , use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of equal length and evaluating \(f\) at the subinterval midpoints. $$ f(x)=1 / x \quad \text { on } \quad[1,9] $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \theta \sqrt[4]{1-\theta^{2}} d \theta $$

Suppose that \(\sum_{k=1}^{n} a_{k}=-5\) and \(\sum_{k=1}^{n} b_{k}=6 .\) Find the values of $$ \begin{array}{ll}{\text { a. } \sum_{k=1}^{n} 3 a_{k}} & {\text { b. } \sum_{k=1}^{n} \frac{b_{k}}{6}} & {\text { c. } \sum_{k=1}^{n}\left(a_{k}+b_{k}\right)} \\ {\text { d. } \sum_{k=1}^{n}\left(a_{k}-b_{k}\right)} & {\text { e. } \sum_{k=1}^{n}\left(b_{k}-2 a_{k}\right)}\end{array} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-\pi / 2}^{\pi / 2}\left(8 y^{2}+\sin y\right) d y $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-3}^{3} \sqrt{9-x^{2}} d x $$

In Exercises \(15-18\) , use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of equal length and evaluating \(f\) at the subinterval midpoints. $$ f(t)=(1 / 2)+\sin ^{2} \pi t \quad \text { on } \quad[0,2] $$

Suppose that \(\sum_{k=1}^{n} a_{k}=0\) and \(\sum_{k=1}^{n} b_{k}=1 .\) Find the values of $$ \begin{array}{ll}{\text { a. } \sum_{k=1}^{n} 8 a_{k}} & {\text { b. } \sum_{k=1}^{n} 250 b_{k}} \\ {\text { c. } \sum_{k=1}^{n}\left(a_{k}+1\right)} & {\text { d. } \sum_{k=1}^{n}\left(b_{k}-1\right)}\end{array} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-\pi / 3}^{-\pi / 4}\left(4 \sec ^{2} t+\frac{\pi}{t^{2}}\right) d t $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-4}^{0} \sqrt{16-x^{2}} d x $$

In Exercises \(15-18\) , use a finite sum to estimate the average value of \(f\) on the given interval by partitioning the interval into four subintervals of equal length and evaluating \(f\) at the subinterval midpoints. $$ f(t)=1-\left(\cos \frac{\pi t}{4}\right)^{4} \quad \text { on } \quad[0,4] $$

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{0}^{\pi} 5(5-4 \cos t)^{1 / 4} \sin t d t $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int 3 y \sqrt{7-3 y^{2}} d y $$

Evaluate the sums in Exercises \(19-28\). $$ \begin{array}{lll}{\text { a. } \sum_{k=1}^{10} k} & {\text { b. } \sum_{k=1}^{10} k^{2}} & {\text { c. } \sum_{k=1}^{10} k^{3}}\end{array} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{1}^{-1}(r+1)^{2} d r $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-2}^{1}|x| d x $$

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{4 y d y}{\sqrt{2 y^{2}+1}} $$

Evaluate the sums in Exercises \(19-28\). $$ \begin{array}{llll}{\text { a. } \sum_{k=1}^{13} k} & {\text { b. } \sum_{k=1}^{13} k^{2}} & {\text { c. } \sum_{k=1}^{13} k^{3}}\end{array} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-\sqrt{3}}^{\sqrt{3}}(t+1)\left(t^{2}+4\right) d t $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-1}^{1}(1-|x|) d x $$

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{0}^{1}\left(4 y-y^{2}+4 y^{3}+1\right)^{-2 / 3}\left(12 y^{2}-2 y+4\right) d y $$

Evaluate the sums in Exercises \(19-28\). $$ \sum_{k=1}^{7}(-2 k) $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{\sqrt{2}}^{1}\left(\frac{u^{7}}{2}-\frac{1}{u^{5}}\right) d u $$

In Exercises \(15-22,\) graph the integrands and use areas to evaluate the integrals. $$ \int_{-1}^{1}(2-|x|) d x $$

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

Evaluate the sums in Exercises \(19-28\). $$ \sum_{k=1}^{5} \frac{\pi k}{15} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{1 / 2}^{1}\left(\frac{1}{v^{3}}-\frac{1}{v^{4}}\right) d v $$