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Chapter 7: Problem 13
Evaluate the following integrals. $$\int \frac{e^{x}}{e^{x}+1} d x$$
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An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), \(f(x)=e^{-a x^{2}}.\) a. Graph the Gaussian for \(a=0.5,1,\) and 2. b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}},\) compute the area under the curves in part (a). c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x,\) where \(a>0, b,\) and \(c\) are real numbers.
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.
\(A\) powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function \(f(t),\) the Laplace transform is a new function \(F(s)\) defined by $$ F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t $$ where we assume that s is a positive real number. For example, to find the Laplace transform of \(f(t)=e^{-t},\) the following improper integral is evaluated: $$ F(s)=\int_{0}^{\infty} e^{-s t} e^{-t} d t=\int_{0}^{\infty} e^{-(s+1) t} d t=\frac{1}{s+1} $$ Verify the following Laplace transforms, where a is a real number. $$f(t)=e^{a t} \longrightarrow F(s)=\frac{1}{s-a}$$
Suppose \(f\) is positive and its first two derivatives are continuous on \([a, b] .\) If \(f^{\prime \prime}\) is positive on \([a, b],\) then is a Trapezoid Rule estimate of \(\int_{a}^{b} f(x) d x\) an underestimate or overestimate of the integral? Justify your answer using Theorem 2 and an illustration.
Evaluate the following integrals or state that they diverge. $$\int_{0}^{\pi / 2} \sec \theta d \theta$$
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