91Ó°ÊÓ

The minimum velocity required for an object to escape Earth's gravitational pull is obtained from the solution of the equation \(\int v d v=-G M \int \frac{1}{y^{2}} d y\) where \(v\) is the velocity of the object projected from Earth, \(y\) is the distance from the center of Earth, \(G\) is the gravitational constant, and \(M\) is the mass of Earth. Show that \(v\) and \(y\) are related by the equation \(v^{2}=v_{0}^{2}+2 G M\left(\frac{1}{y}-\frac{1}{R}\right)\) where \(v_{0}\) is the initial velocity of the object and \(R\) is the radius of Earth.

Evaluate, if possible, the integral \(\int_{0}^{2} \llbracket x \rrbracket d x\).

\(x(t)=t^{3}-6 t^{2}+9 t-2, \quad 0 \leq t \leq 5\) (a) Find the velocity and acceleration of the particle. (b) Find the open \(t\) -intervals on which the particle is moving to the right. (c) Find the velocity of the particle when the acceleration is \(0 .\)

A particle moves along the \(x\) -axis at a velocity of \(v(t)=1 / \sqrt{t}\) \(t>0\). At time \(t=1\), its position is \(x=4\). Find the acceleration and position functions for the particle.

If \(p(x)\) is a polynomial function, then \(p\) has exactly one antiderivative whose graph contains the origin.

Evaluate the integral using the properties of even and odd functions as an aid. $$ \int_{-2}^{2} x^{2}\left(x^{2}+1\right) d x $$

Use the symmetry of the graphs of the sine and cosine functions as an aid in evaluating each definite integral. (a) \(\int_{-\pi / 4}^{\pi / 4} \sin x d x\) (b) \(\int_{-\pi / 4}^{\pi / 4} \cos x d x\) (c) \(\int_{-\pi / 2}^{\pi / 2} \cos x d x\) (d) \(\int_{-\pi / 2}^{\pi / 2} \sin x \cos x d x\)

Write the integral as the sum of the integral of an odd function and the integral of an even function. Use this simplification to evaluate the integral. $$ \int_{-4}^{4}\left(x^{3}+6 x^{2}-2 x-3\right) d x $$

The oscillating current in an electrical circuit is \(I=2 \sin (60 \pi t)+\cos (120 \pi t)\) where \(I\) is measured in amperes and \(t\) is measured in seconds. Find the average current for each time interval. (a) \(0 \leq t \leq \frac{1}{60}\) (b) \(0 \leq t \leq \frac{1}{240}\) (c) \(0 \leq t \leq \frac{1}{30}\)