91Ó°ÊÓ

Are concerned with a tank filled to the top with water. Its shape is obtained by rotating about the \(y\) -axis the region that is bounded above by the horizontal line \(y=10\) \(\left(1-1 / e^{2}\right),\) on the left by the \(y\) -axis, and on the right by the graph of \(y=10\left(1-\exp \left(-x^{2} / 2\right)\right) .\) Both \(x\) and \(y\) are measured in feet. A pump floats on the surface of the water and pumps water to the top, at which point the water runs off. How much work is done in pumping out half the water in the tank?

Is the average value of \(\cos (x)\) for \(0 \leq x \leq \pi / 4\) equal to the reciprocal of the average value of \(1 / \cos (x)\) over the same \(x\) -interval?

Use Simpson's Rule to calculate the arc length of the graph of \(y=\sin (x), 0 \leq x \leq 2 \pi,\) to four decimal places of accuracy.

Calculate the volume of the solid obtained when the triangle with vertices (2,5),(6,1),(4,4) is rotated about the line \(x=-3\).

Calculate the volume obtained when the region outside the square \(\\{(x, y):|x|<1,|y|<1\\}\) and inside the circle \(\\{(x, y)\) : \(\left.x^{2}+y^{2} \leq 4\right\\}\) is rotated about the line \(y=-3\)

An open cylindrical beaker with circular base has height \(L\) and radius \(r\). It is partially filled with a volume \(V\) of a fluid. Consider the parameters \(L, r,\) and \(V\) to be constant. The axis of symmetry of the beaker is along the positive \(y\) -axis and one diameter of its base is along the \(x\) -axis. When the tank is revolved about the \(y\) -axis with angular speed \(\omega\), the surface of the fluid assumes a shape that is the paraboloid of revolution that results when the curve $$ y=h+\omega^{2} x^{2} /(2 g), \quad 0 \leq x \leq r $$ is revolved about the \(y\) -axis. This formula is valid for angular speeds at which the surface of the fluid has not yet touched the base or the mouth of the beaker. The number \(h=h(\omega)\) is in the interval \(\left[0, V /\left(\pi r^{2}\right)\right]\) and depends on \(\omega\). (When \(\omega=0,\) then \(h=V /\left(\pi r^{2}\right)\). As \(\omega\) increases, \(h\) decreases.) a. Find a formula for \(h(\omega)\). b. At what value \(\omega_{S}\) of \(\omega\) does spilling begin, assuming $$ \text { that } h(\omega)>0 \text { for } \omega>\omega_{S} ? $$ c. At what value \(\omega_{B}\) of \(\omega\) does the surface touch the bottom of the beaker, assuming that spilling does not $$ \text { occur for } \omega<\omega_{B} ? $$ d. As \(\omega\) increases, does the surface of the fluid touch the bottom of the beaker or the mouth of the beaker first?

Suppose an object of mass \(m\) is propelled upwards from the surface of the earth with initial velocity \(v_{0}\). Suppose that the (downward) force of air resistance \(R(v)\) is proportional to the square of the speed: \(R(v)=-k \cdot v^{2},\) where \(k\) is a positive constant that carries the units of mass/ length. (This is the quadratic drag law.) Solve the initial value problem for motion: $$ m \frac{d v}{d t}=-k v^{2}-m g, \quad v(0)=v_{0} $$

The rate of elimination of alcohol from the bloodstream is proportional to the amount \(A\) that is present. That is, $$ \frac{d A}{d t}=-\frac{1}{k} A $$ where \(k\) is a time constant that depends on the drug and the individual. If \(k\) is \(1 / 2\) hour for a certain person, how long will it take for his blood alcohol content to reduce from \(0.12 \%\) to \(0.06 \% ?\)

Solve the given initial value problem for \(y(x)\). Determine the value of \(y(2)\). $$ y^{2} \cdot d y / d x=(1+y) /(1+2 x) \quad y(0)=0 $$