Problem 67
This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of \(F\) is the set of all real numbers s such that the improper integral converges. Find the Laplace transform \(F\) of each of the following functions and give the domain of \(F\).\(f(x)=e^{a x}\)
Problem 67
Find the volume of the solid generated by revolving the region bounded by the curve \(y=4 \cos x\) and the \(x\) -axis, \(\frac{\pi}{2} \leq x \leq \frac{3 \pi}{2}\), about the \(x\) -axis. (Express the answer in exact form.)
Problem 68
Find the area of the region bounded by the graphs of the equations \(y=\cos ^{2} x, y=\sin ^{2} x, x=-\frac{\pi}{4}\), and \(x=\frac{\pi}{4}\).
Problem 68
Use the formula for arc length to show that the circumference of the circle \(x^{2}+y^{2}=1\) is \(2 \pi\).
Problem 68
Find the volume of the solid generated by revolving the region in the first quadrant bounded by \(y=e^{x}\) and the \(x\) -axis, from \(x=0\) to \(x=\ln (7)\), about the \(y\) -axis. (Express the answer in exact form.)
Problem 69
A function is a probability density function if it satisfies the following definition: \(\int_{-\infty}^{\infty} f(t) d t=1\). The probability that a random variable \(x\) lies between \(a\) and \(b\) is given by \(P(a \leq x \leq b)=\int_{a}^{b} f(t) d t\)Show that \(f(x)=\left\\{\begin{array}{c}0 \text { if } x<0 \\ 7 e^{-7 x} \text { if } x \geq 0\end{array}\right.\) is a probability density function.
Problem 69
A particle moves in a straight line with the velocity function \(v(t)=\sin (\omega t) \cos ^{2}(\omega t) .\) Find its position function \(x=f(t)\) if \(f(0)=0 .\)
Problem 70
Find the average value of the function \(f(x)=\sin ^{2} x \cos ^{3} x\) over the interval \([-\pi, \pi]\).
Problem 71
For the following exercises, solve the differential equations. \(\frac{d y}{d x}=\sin ^{2} x\). The curve passes through point \((0,0)\).
Problem 71
For the following problems, use the substitutions \(\tan \left(\frac{x}{2}\right)=t, d x=\frac{2}{1+t^{2}} d t, \sin x=\frac{2 t}{1+t^{2}}\), and \(\cos x=\frac{1-t^{2}}{1+t^{2}}\). \(\int \frac{d x}{3-5 \sin x}\)
