Problem 82
Magnetic field due to current in a straight wire A long straight wire of length \(2 L\) on the \(y\) -axis carries a current \(I\). According to the Biot- Savart Law, the magnitude of the magnetic field due to the current at a point \((a, 0)\) is given by $$ B(a)=\frac{\mu_{0} I}{4 \pi} \int_{-L}^{L} \frac{\sin \theta}{r^{2}} d y $$ where \(\mu_{0}\) is a physical constant, \(a>0,\) and \(\theta, r,\) and \(y\) are related as shown in the figure. a. Show that the magnitude of the magnetic field at \((a, 0)\) is $$ B(a)=\frac{\mu_{0} I L}{2 \pi a \sqrt{a^{2}+L^{2}}} $$ b. What is the magnitude of the magnetic field at \((a, 0)\) due to an infinitely long wire \((L \rightarrow \infty) ?\)
Problem 82
Determine whether the following integrals converge or diverge. $$\int_{1}^{\infty} \frac{1}{e^{x}\left(1+x^{2}\right)} d x$$
Problem 83
Determine whether the following integrals converge or diverge. $$\int_{1}^{\infty} \frac{2+\cos x}{\sqrt{x}} d x$$
Problem 83
Evaluate \(\int_{0}^{\pi / 4} \ln (1+\tan x) d x\) using the following steps. a. If \(f\) is integrable on \([0, b],\) use substitution to show that $$\int_{0}^{b} f(x) d x=\int_{0}^{b / 2}(f(x)+f(b-x)) d x$$ b. Use part (a) and the identity tan \((\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}\) to evaluate \(\int_{0}^{\pi / 4} \ln (1+\tan x) d x\) (Source: The College Mathematics Journal, \(33,4,\) Sep 2004 )
Problem 83
Evaluate the following integrals. $$\int e^{x} \sin ^{998}\left(e^{x}\right) \cos ^{3}\left(e^{x}\right) d x$$
Problem 84
Evaluate the following integrals. $$\int \frac{\tan \theta+\tan ^{3} \theta}{(1+\tan \theta)^{50}} d \theta$$
Problem 84
Determine whether the following integrals converge or diverge. $$\int_{1}^{\infty} \frac{2+\cos x}{x^{2}} d x$$
Problem 84
Evaluate \(\int \cos (\ln x) d x\) two different ways: a. Use tables after first using the substitution \(u=\ln x\). B. Use integration by parts twice to verify your answer to part (a)
Problem 84
Maximum path length of a projectile (Adapted from Putnam Exam 1940) A projectile is launched from the ground with an initial speed \(V\) at an angle \(\theta\) from the horizontal. Assume the \(x\) -axis is the horizontal ground and \(y\) is the height above the ground. Neglecting air resistance and letting \(g\) be the acceleration due to gravity, it can be shown that the trajectory of the projectile is given by $$ \begin{array}{l} y=-\frac{1}{2} k x^{2}+y_{\max }, \quad \text { where } k=\frac{g}{(V \cos \theta)^{2}} \\ \text { and } y_{\max }=\frac{(V \sin \theta)^{2}}{2 g} \end{array} $$ a. Note that the high point of the trajectory occurs at \(\left(0, y_{\max }\right)\) If the projectile is on the ground at \((-a, 0)\) and \((a, 0)\) what is \(a ?\) b. Show that the length of the trajectory (arc length) is $$ 2 \int_{0}^{a} \sqrt{1+k^{2} x^{2}} d x $$ c. Evaluate the arc length integral and express your result in terms of \(V, g,\) and \(\theta\) d. For a fixed value of \(V\) and \(g,\) show that the launch angle \(\theta\) that maximizes the length of the trajectory satisfies \((\sin \theta) \ln (\sec \theta+\tan \theta)=1\) e. Use a graphing utility to approximate the optimal launch angle.
Problem 85
Determine whether the following integrals converge or diverge. $$\int_{0}^{1} \frac{d x}{\sqrt{x^{1 / 3}+x}}$$
