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Chapter 7: Problem 13
Evaluate the following integrals. $$\int \frac{d x}{\left(16-x^{2}\right)^{1 / 2}}$$
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The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d x}{e^{x}+e^{2 x}}$$
Suppose that the rate at which a company extracts oil is given by \(r(t)=r_{0} e^{-k t},\) where \(r_{0}=10^{7}\) barrels \(/ \mathrm{yr}\) and \(k=0.005 \mathrm{yr}^{-1} .\) Suppose also the estimate of the total oil reserve is \(2 \times 10^{9}\) barrels. If the extraction continues indefinitely, will the reserve be exhausted?
Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\\\&\sin m x \cos n x=\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\\\&\cos m x \cos n x=\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\end{aligned}$$ $$\int \cos x \cos 2 x d x$$
Evaluate \(\int \frac{d y}{y(\sqrt{a}-\sqrt{y})},\) for \(a > 0\). (Hint: Use the substitution \(u=\sqrt{y}\) followed by partial fractions.)
The cycloid is the curve traced by a point on the rim of a rolling wheel. Imagine a wire shaped like an inverted cycloid (see figure). A bead sliding down this wire without friction has some remarkable properties. Among all wire shapes, the cycloid is the shape that produces the fastest descent time. It can be shown that the descent time between any two points \(0 \leq a \leq b \leq \pi\) on the curve is $$\text { descent time }=\int_{a}^{b} \sqrt{\frac{1-\cos t}{g(\cos a-\cos t)}} d t$$ where \(g\) is the acceleration due to gravity, \(t=0\) corresponds to the top of the wire, and \(t=\pi\) corresponds to the lowest point on the wire. a. Find the descent time on the interval \([a, b]\) by making the substitution \(u=\cos t\) b. Show that when \(b=\pi\), the descent time is the same for all values of \(a ;\) that is, the descent time to the bottom of the wire is the same for all starting points.
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