Problem 62
A particle moving along a straight line has a velocity of \(v(t)=t^{2} e^{-t}\) after \(t \mathrm{sec}\). How far does it travel in the first 2 sec? (Assume the units are in feet and express the answer in exact form.)
Problem 63
The velocity of a particle moving along a line is a function of time given by \(v(t)=\frac{88 t^{2}}{t^{2}+1} .\) Find the distance that the particle has traveled after \(t=5 \mathrm{sec}\).
Problem 63
For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible. \(\int_{0}^{4 \pi} \cos (x / 2) \sin (x / 2) d x\)
Problem 63
Find the area under the graph of \(y=\sec ^{3} x\) from \(x=0\) to \(x=1\). (Round the answer to two significant digits.)
Problem 63
Find the volume of the solid generated by revolving about the \(y\) -axis the region under the curve \(y=6 e^{-2 x}\) in the first quadrant.
Problem 63
Given that we know the Fundamental Theorem of Calculus, why would we want to develop numerical methods for definite integrals?
Problem 64
Find the area between \(y=(x-2) e^{x}\) and the \(x\) -axis from \(x=2\) to \(x=5\). (Express the answer in exact form.)
Problem 64
For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible. \(\int_{\pi / 6}^{\pi / 3} \frac{\cos ^{3} x}{\sqrt{\sin x}} d x\) (Round this answer to three decimal places.)
Problem 64
Solve the initial-value problem for \(x\) as a function of \(t\). \(\left(t^{2}-7 t+12\right) \frac{d x}{d t}=1,(t>4, x(5)=0)\)
Problem 64
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 \(F\).\(f(x)=1\)
