Let $g$ be the function given by $g ( x ) = \frac { 1 } { \sqrt { x } }$.
(a) Find the average value of $g$ on the closed interval $[ 1,4 ]$.
(b) Let $S$ be the solid generated when the region bounded by the graph of $y = g ( x )$, the vertical lines $x = 1$ and $x = 4$, and the $x$-axis is revolved about the $x$-axis. Find the volume of $S$.
(c) For the solid $S$, given in part (b), find the average value of the areas of the cross sections perpendicular to the $x$-axis.
(d) The average value of a function $f$ on the unbounded interval $[ a , \infty )$ is defined to be $\lim _ { b \rightarrow \infty } \left[ \frac { \int _ { a } ^ { b } f ( x ) d x } { b - a } \right]$. Show that the improper integral $\int _ { 4 } ^ { \infty } g ( x ) d x$ is divergent, but the average value of $g$ on the interval $[ 4 , \infty )$ is finite.

Let $f(x) = e^{2x}$. Let $R$ be the region in the first quadrant bounded by the graph of $f$, the coordinate axes, and the vertical line $x = k$, where $k > 0$. The region $R$ is shown in the figure above.
(a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of $R$ in terms of $k$.
(b) The region $R$ is rotated about the $x$-axis to form a solid. Find the volume, $V$, of the solid in terms of $k$.
(c) The volume $V$, found in part (b), changes as $k$ changes. If $\frac{dk}{dt} = \frac{1}{3}$, determine $\frac{dV}{dt}$ when $k = \frac{1}{2}$.

We consider the function $f$ defined on $\mathbb { R }$ by:
$$f ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$$
Let A be a point on $\mathscr { C }$ with positive abscissa $a$. The rotation around the x-axis applied to the part of $\mathscr { C }$ bounded by points I and A generates a surface modeling the flute container, taking 1 cm as the unit.
The real number $a$ being strictly positive, we admit that the volume $V ( a )$ of this solid in $\mathrm { cm } ^ { 3 }$ is given by the formula:
$$V ( a ) = \pi \int _ { 0 } ^ { a } ( f ( x ) ) ^ { 2 } \mathrm {~d} x$$

1. Verify, for all real numbers $x \geqslant 0$, the equality: $$( f ( x ) ) ^ { 2 } = 4 \left( \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 } + \frac { - \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } } \right) .$$
2. Determine a primitive on $\mathbb { R }$ of each of the functions: $$g : x \longmapsto \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 } \quad \text { and } \quad h : x \longmapsto \frac { - \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } }$$
3. Deduce that for all real $a > 0$: $$V ( a ) = 4 \pi \left[ \ln \left( \frac { \mathrm { e } ^ { a } + 1 } { 2 } \right) + \frac { 1 } { \mathrm { e } ^ { a } + 1 } - \frac { 1 } { 2 } \right] .$$
4. Using a calculator, determine an approximate value of $a$ to 0.1, knowing that a flute must contain $12.5 \mathrm { cL }$, that is $125 \mathrm {~cm} ^ { 3 }$. No justification is required.

Find the volume of the solid obtained when the region bounded by $y = \sqrt{x}$, $y = -x$ and the line $x = 9$ is revolved around the $x$-axis. (It may be useful to draw the specified region.)

The volume of the solid of revolution created by rotating the region enclosed by the two curves $y = \sqrt { x } , y = \sqrt { - x + 10 }$ and the $x$-axis around the $x$-axis is $a \pi$. Find the value of $a$. [3 points]

As shown in the figure, there is a line $l : x - y - 1 = 0$ and a hyperbola $C : x ^ { 2 } - 2 y ^ { 2 } = 1$ with one focus at point $\mathrm { F } ( c , 0 )$ (where $c < 0$).
When the region enclosed by the line $l$ and the hyperbola $C$ is rotated about the $y$-axis, what is the volume of the solid of revolution? [3 points]
(1) $\frac { 5 } { 3 } \pi$
(2) $\frac { 3 } { 2 } \pi$
(3) $\frac { 4 } { 3 } \pi$
(4) $\frac { 7 } { 6 } \pi$
(5) $\pi$

Consider the function $$f ( x ) = \begin{cases} | 5 x ( x + 2 ) | & ( x < 0 ) \\ | 5 x ( x - 2 ) | & ( x \geq 0 ) \end{cases}$$ On the closed interval $[ 0,1 ]$, what is the volume of the solid of revolution generated by rotating the region enclosed by the graph of $y = f ( x )$, the $x$-axis, and the line $x = 1$ about the $x$-axis? [3 points]
(1) $\frac { 65 } { 6 } \pi$
(2) $\frac { 35 } { 3 } \pi$
(3) $\frac { 25 } { 2 } \pi$
(4) $\frac { 40 } { 3 } \pi$
(5) $\frac { 85 } { 6 } \pi$

Consider a twice differentiable function $y ( x )$ in an $x y$ plane which connects two points $A ( - 1,2 )$ and $B ( 1,2 )$. Let $S$ be outer surface area of the cylindrical object created by rotation of the curve $y ( x )$ about the $x$ axis. Answer the following questions.
(1) Prove that the surface area $S$ is given by
$$\begin{aligned} S & = 2 \pi \int _ { - 1 } ^ { 1 } F \left( y , y ^ { \prime } \right) \mathrm { d } x \\ F \left( y , y ^ { \prime } \right) & = y \sqrt { 1 + \left( y ^ { \prime } \right) ^ { 2 } } \end{aligned}$$
where $y ^ { \prime } = \frac { \mathrm { d } y } { \mathrm {~d} x }$.
(2) Let the curve $y ( x )$ satisfy the following Euler-Lagrange equation for arbitrary $x$:
$$\frac { \partial F } { \partial y } - \frac { \mathrm { d } } { \mathrm {~d} x } \frac { \partial F } { \partial y ^ { \prime } } = 0$$
Considering Eq. (2.3) along with $\frac { \mathrm { d} F } { \mathrm {~d} x }$, prove that the following relation holds:
$$F - y ^ { \prime } \frac { \partial F } { \partial y ^ { \prime } } = c$$
Here $c$ is a constant.
(3) Express a differential equation satisfied by the curve $y ( x )$ using $y , y ^ { \prime } , c$.
(4) Represent the curve $y ( x )$ as a function of $x$ and $c$.
Obtain an equation which should be satisfied by the constant $c$.

In the analytic plane; the region bounded by the x-axis, the line $x + y = 2$, and the curve $y = \sqrt { x }$ is rotated $360 ^ { \circ }$ around the x-axis.
What is the volume of the solid of revolution obtained in cubic units?
A) $\frac { \pi } { 2 }$
B) $\frac { 2 \pi } { 3 }$
C) $\frac { 3 \pi } { 4 }$
D) $\frac { 5 \pi } { 6 }$
E) $\frac { 7 \pi } { 6 }$

In the rectangular coordinate plane, the region between the parabola $y = x ^ { 2 }$, the line $x = 1$, and the line $y = 0$ is shown.
What is the volume in cubic units of the solid of revolution obtained by rotating this region $360 ^ { \circ }$ about the line $\mathbf { y = - 1 }$?
A) $\frac { 3 \pi } { 4 }$
B) $\frac { 5 \pi } { 8 }$
C) $\frac { 7 \pi } { 10 }$
D) $\frac { 11 \pi } { 12 }$
E) $\frac { 13 \pi } { 15 }$

In the rectangular coordinate plane, the region between the lines $y = - x + 5$, $y = x + 3$ and the coordinate axes is shown below.
What is the volume of the solid of revolution obtained by rotating this region $360 ^ { \circ }$ about the y-axis?
A) $37 \pi$
B) $38 \pi$
C) $40 \pi$
D) $41 \pi$
E) $42 \pi$