Chapter 6

Given that \(\int_{0}^{a} x^{4} d x=\frac{1}{5} a^{5}\) evaluate the following integrals (a) \(\int_{0}^{2} x^{4} d x\) (b) \(\int_{0}^{1} \frac{x^{4}}{2} d x\) (c) \(\int_{-1}^{1} \frac{x^{4}}{2} d x\) (d) \(\int_{-2}^{0}(x+2)^{4} d x\) (e) \(\int_{-3}^{0}(x+1)^{4} d x\) (f) \(\int_{0}^{2} 2(x-2)^{4} d x\).

Compute the indefinite integrals. $$ \int 3 e^{-x} d x $$

In Problems 59-63, verify each inequality without evaluating the integrals. $$ \int_{0}^{1} x d x \geq \int_{0}^{1} x^{2} d x $$

Compute the indefinite integrals. $$ \int 2 e^{-x / 3} d x $$

In Problems , verify each inequality without evaluating the integrals. $$ \int_{2}^{4} x d x \leq \int_{2}^{4} x^{2} d x $$

In Problems , verify each inequality without evaluating the integrals. $$ 0 \leq \int_{0}^{9} \sqrt{x} d x \leq 27 $$

Find the length of the curve $$ 3 y^{2}=4 x^{3} $$ from \(x=0\) to \(x=1\).

Compute the indefinite integrals. $$ \int e^{x}\left(1-e^{-x}\right) d x $$

Find the length of the curve $$ y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}} $$ from \(x=1\) to \(x=3\).

Compute the indefinite integrals. $$ \int \cos (3 x) d x $$

Find the length of the curve $$ y=\frac{x^{3}}{6}+\frac{1}{2 x} $$ from \(x=2\) to \(x=4\).

Compute the indefinite integrals. $$ \int \sin \frac{x}{3} d x $$

$$ \text { Find the value of } a \geq 0 \text { that maximizes } \int_{0}^{a}\left(4-x^{2}\right) d x \text { . } $$

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=x^{2},-1 \leq x \leq 1\)

Compute the indefinite integrals. $$ \int \cos (3 x) d x $$

$$ \text { Find the value of } a \in[0,2 \pi] \text { that maximizes } \int_{0}^{a} \cos x d x \text { . } $$

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=x^{2}+1,-1 \leq x \leq 1\)

Compute the indefinite integrals. $$ \int \cos (2+x) d x $$

$$ \text { Find } a \in(0,2 \pi] \text { such that } \int_{0}^{a} \sin x d x=0 $$

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=e^{-x}, 0 \leq x \leq 1\)

Compute the indefinite integrals. $$ \int \sin (2 x-1) d x $$

$$ \text { Find } a>1 \text { such that } \int_{1}^{a}(x-3)^{3} d x=0 $$

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=\frac{1}{x}, 1 \leq x \leq 2\)

Compute the indefinite integrals. $$ \int \cos (2 x+1) d x $$

$$ \text { Find } a>0 \text { such that } \int_{0}^{a}(1-x) d x=0 $$

Find the length of the quarter-circle $$ y=\sqrt{1-x^{2}} $$ for \(0 \leq x \leq 1\), by each of the following methods: (a) a formula from geometry (b) the integral formula from Subsection \(6.3 .6\)

Compute the indefinite integrals. $$ \int \frac{\sin x}{1-\sin ^{2} x} d x $$

Total Rainfall A rain gauge is set up to measure the amount of rainfall occurring in \(1 \mathrm{hr}\) on the UCLA campus (the readout from the rain gauge is in \(\mathrm{mm} / \mathrm{hr}\) ). Assume that the following data is collected in a 6 hour window. $$ \begin{array}{c|c} \hline \text { Time, } t & \text { Rainfall rate, } \boldsymbol{r}(\boldsymbol{t}) \text { in } \mathrm{mm} / \mathrm{hr} \\ \hline 0 & 1 \\ 1 & 2 \\ 2 & 3 \\ 3 & 1 \\ 4 & 1 \\ 5 & 0 \\ 6 & 0 \\ \hline \end{array} $$

A cable that hangs between two poles at \(x=-M\) and \(x=M\) takes the shape of a catenary, with equation $$ y=\frac{1}{2 a}\left(e^{a x}+e^{-a x}\right) $$ where \(a\) is a positive constant. Compute the length of the cable when \(a=1\) and \(M=\ln 3\).

Compute the indefinite integrals. $$ \int \frac{\cos x}{1-\cos ^{2} x} d x $$

You are measuring the ability of an antibiotic to kill harmful bacteria. You measure the rate at which the antibiotic kills bacteria (i.e., number of bacteria killed in one hour); this is called the mortality rate. You measure the following data for the number of bacteria killed in a 12 hour time period starting at \(t=0\), and ending at \(t=12\). $$ \begin{array}{c|c} \hline \text { Time, } t & \text { Mortality rate, per hour } \boldsymbol{m}(\boldsymbol{t}) \\ \hline 0 & 20 \\ 1 & 300 \\ 2 & 350 \\ 3 & 400 \\ 4 & 500 \\ 5 & 450 \\ 6 & 410 \\ 7 & 350 \\ 8 & 320 \\ 9 & 300 \\ 10 & 200 \\ 11 & 100 \\ 12 & 110 \\ \hline \end{array} $$ (a) Use six even subintervals to approximate the total number of deaths between \(t=0\) and \(t=6\) and evaluate this sum using the data in the table. (b) Use six even subintervals to approximate the total number of deaths between \(t=0\) and \(t=12\) and evaluate this sum using the data in the table. (c) Use four even subintervals to approximate the total number of deaths between \(t=4\) and \(t=12\) and evaluate this sum using the data in the table.

Show that if $$ f(x)=\frac{e^{x}+e^{-x}}{2} $$ then the length of the curve \(f(x)\) between \(x=0\) and \(x=a\) for any \(a>0\) is given by \(f^{\prime}(a)\).

Compute the indefinite integrals. $$ \int \cos x \sin x d x $$

Compute the indefinite integrals. $$ \int\left(\cos ^{2} x-\sin ^{2} x\right) d x $$

Compute the indefinite integrals. $$ \int\left(\cos x+\cos ^{2} x\right) d x $$

Compute the indefinite integrals. $$ \int\left(\sin x-\sin ^{2} x\right) d x $$

Compute the indefinite integrals. $$ \int \frac{4}{1+x^{2}} d x $$

Compute the indefinite integrals. $$ \int\left(\frac{x^{2}}{1+x^{2}}\right) d x $$

Compute the indefinite integrals. $$ \int \frac{1}{\sqrt{1-x^{2}}} d x $$

Compute the indefinite integrals. $$ \int \frac{5}{\sqrt{1-x^{2}}} d x $$

Compute the indefinite integrals. $$ \int \frac{1}{x+2} d x $$

Compute the indefinite integrals. $$ \int \frac{1}{x-3} d x $$

Compute the indefinite integrals. $$ \int \frac{2 x-1}{3 x} d x $$

Compute the indefinite integrals. $$ \int \frac{2 x+5}{x} d x $$

Compute the indefinite integrals. $$ \int \frac{1}{2 x+1} d x $$

Compute the indefinite integrals. $$ \int \frac{1}{3 x-3} d x $$

Compute the indefinite integrals. $$ \int \frac{1}{x^{2}+4} d x $$

Compute the indefinite integrals. $$ \int \frac{1}{x^{2}} d x $$

Compute the indefinite integrals. $$ \int \frac{2 x^{2}}{x^{2}+1} d x $$

Compute the indefinite integrals. $$ \int \frac{2 x^{2}}{4+x^{2}} d x $$