A company designs spinning toys using the family of functions $y = c x \sqrt { 4 - x ^ { 2 } }$, where $c$ is a positive constant. The figure above shows the region in the first quadrant bounded by the $x$-axis and the graph of $y = c x \sqrt { 4 - x ^ { 2 } }$, for some $c$. Each spinning toy is in the shape of the solid generated when such a region is revolved about the $x$-axis. Both $x$ and $y$ are measured in inches.
(a) Find the area of the region in the first quadrant bounded by the $x$-axis and the graph of $y = c x \sqrt { 4 - x ^ { 2 } }$ for $c = 6$.
(b) It is known that, for $y = c x \sqrt { 4 - x ^ { 2 } } , \frac { d y } { d x } = \frac { c \left( 4 - 2 x ^ { 2 } \right) } { \sqrt { 4 - x ^ { 2 } } }$. For a particular spinning toy, the radius of the largest cross-sectional circular slice is 1.2 inches. What is the value of $c$ for this spinning toy?
(c) For another spinning toy, the volume is $2 \pi$ cubic inches. What is the value of $c$ for this spinning toy?

Exercise 1 (5 points)
The Delmas chocolate factory decides to market new confectionery: chocolate drops in the shape of a water droplet. To do this, it must manufacture custom moulds that must meet the following constraint: for this range of sweets to be profitable, the chocolate factory must be able to produce at least 80 with 1 litre of liquid chocolate paste.

Part A: modelling by a function

The half-perimeter of the upper face of the drop will be modelled by a portion of the curve of the function $f$ defined on $]0;+\infty[$ by: $$f(x) = \frac{x^2 - 2x - 2 - 3\ln x}{x}.$$

1. Let $\varphi$ be the function defined on $]0;+\infty[$ by: $$\varphi(x) = x^2 - 1 + 3\ln x.$$ a. Calculate $\varphi(1)$ and the limit of $\varphi$ at 0. b. Study the variations of $\varphi$ on $]0;+\infty[$. Deduce the sign of $\varphi(x)$ according to the values of $x$.
   
2. a. Calculate the limits of $f$ at the boundaries of its domain of definition. b. Show that on $]0;+\infty[$: $f'(x) = \dfrac{\varphi(x)}{x^2}$. Deduce the variation table of $f$. c. Prove that the equation $f(x) = 0$ has a unique solution $\alpha$ on $]0;1]$. Determine using a calculator an approximate value of $\alpha$ to $10^{-2}$ near. It is admitted that the equation $f(x) = 0$ also has a unique solution $\beta$ on $[1;+\infty[$ with $\beta \approx 3.61$ to $10^{-2}$ near. d. Let $F$ be the function defined on $]0;+\infty[$ by: $$F(x) = \frac{1}{2}x^2 - 2x - 2\ln x - \frac{3}{2}(\ln x)^2.$$ Show that $F$ is an antiderivative of $f$ on $]0;+\infty[$.

Part B: solving the problem

In this part, calculations will be performed with the approximate values to $10^{-2}$ near of $\alpha$ and $\beta$ from Part A. To obtain the shape of the droplet, we consider the representative curve $C$ of the function $f$ restricted to the interval $[\alpha;\beta]$ as well as its reflection $C'$ with respect to the horizontal axis. The two curves $C$ and $C'$ delimit the upper face of the drop. For aesthetic reasons, the chocolatier would like his drops to have a thickness of $0.5$ cm. Under these conditions, would the profitability constraint be respected?

By rotating the cross-section figure around the x-axis, we obtain a model of the light bulb. We decompose it into three parts. We recall that:

• the volume of a cylinder is given by the formula $\pi r^2 h$ where $r$ is the radius of the base disk and $h$ is the height;
• the volume of a sphere of radius $r$ is given by the formula $\frac{4}{3}\pi r^3$.

We also admit that, for every real number $x$ in the interval $[0;4]$, $f(x) = 2 - \cos\left(\frac{\pi}{4}x\right)$.
The points are $\mathrm{A}(-1;1)$, $\mathrm{B}(0;1)$, $\mathrm{C}(4;3)$, $\mathrm{D}(7;0)$, $\mathrm{E}(4;-3)$, $\mathrm{F}(0;-1)$, $\mathrm{G}(-1;-1)$.

1. Calculate the volume of the cylinder with cross-section the rectangle $ABFG$.
2. Calculate the volume of the hemisphere with cross-section the half-disk with diameter $[CE]$.
3. To approximate the volume of the solid with cross-section the shaded region BCEF, we divide the segment $[OO']$ into $n$ segments of equal length $\frac{4}{n}$ then we construct $n$ cylinders of equal height $\frac{4}{n}$. a. Special case: in this question only we choose $n = 5$. Calculate the volume of the third cylinder, shaded in the figures, then give its value rounded to $10^{-2}$. b. General case: in this question, $n$ denotes any non-zero natural number. We approximate the volume of the solid with cross-section BCEF by the sum of the volumes of the $n$ cylinders thus created by choosing a sufficiently large value of $n$. Copy and complete the following algorithm so that at the end of its execution, the variable $V$ contains the sum of the volumes of the $n$ cylinders created when $n$ is entered. \begin{verbatim} $V \leftarrow 0$ For $k$ going from...to ... : $\mid V \leftarrow \ldots$ Fin For \end{verbatim}

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.

A company produces cylindrical cans to store food. The technical standard requires that the ratio between the height and the diameter of the base of the can must be equal to 2. A can that meets this standard has a volume of $V = 500\pi \text{ cm}^3$.
What is the height, in centimeters, of this can?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25

To find the volume of a cave, we fit $\mathrm { X } , \mathrm { Y }$ and Z axes such that the base of the cave is in the XY-plane and the vertical direction is parallel to the Z-axis. The base is the region in the XY-plane bounded by the parabola $y ^ { 2 } = 1 - x$ and the Y-axis. Each cross-section of the cave perpendicular to the X-axis is a square.
(a) Show how to write a definite integral that will calculate the volume of this cave.
(b) Evaluate this definite integral. Is it possible to evaluate it without using a formula for indefinite integrals?

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.)

When a solid of revolution is created by rotating the figure enclosed by the curve $y = a \left( 1 - x ^ { 2 } \right)$ and the $x$-axis around the $y$-axis, and the volume of the solid of revolution is $16 \pi$, find the positive value of $a$. [3 points]

The region enclosed by the curve $y = \frac { 1 } { 4 } x ^ { 2 }$ and the line $y = 4$ is rotated around the $y$-axis. If the volume of the solid of revolution is $k \pi$, find the value of the constant $k$. [3 points]

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) $\frac { 85 } { 6 } \pi$

As shown in the figure, there is a solid figure with base formed by the curve $y = \sqrt { x } + 1$, the $x$-axis, the $y$-axis, and the line $x = 1$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is always a square, what is the volume of this solid figure? [3 points]
(1) $\frac { 7 } { 3 }$
(2) $\frac { 5 } { 2 }$
(3) $\frac { 8 } { 3 }$
(4) $\frac { 17 } { 6 }$
(5) 3

As shown in the figure, for a positive number $k$, the region enclosed by the curve $y = \sqrt { \frac { e ^ { x } } { e ^ { x } + 1 } }$, the $x$-axis, the $y$-axis, and the line $x = k$ is the base of a solid figure. When the cross-section perpendicular to the $x$-axis is always a square and the volume is $\ln 7$, what is the value of $k$? [3 points]
(1) $\ln 11$
(2) $\ln 13$
(3) $\ln 15$
(4) $\ln 17$
(5) $\ln 19$

As shown in the figure, there is a solid figure with base formed by the curve $y = \sqrt{(1-2x)\cos x}$ ($\frac{3}{4}\pi \leq x \leq \frac{5}{4}\pi$) and the $x$-axis and the two lines $x = \frac{3}{4}\pi$ and $x = \frac{5}{4}\pi$. When this solid figure is cut by a plane perpendicular to the $x$-axis, all cross-sections are squares. Find the volume of this solid figure. [3 points]
(1) $\sqrt{2}\pi - \sqrt{2}$
(2) $\sqrt{2}\pi - 1$
(3) $2\sqrt{2}\pi - \sqrt{2}$
(4) $2\sqrt{2}\pi - 1$
(5) $2\sqrt{2}\pi$

As shown in the figure, a solid figure has as its base the region enclosed by the curve $y = \sqrt{\frac{x+1}{x(x + \ln x)}}$, the $x$-axis, and the two lines $x = 1$ and $x = e$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is a square, what is the volume of this solid figure? [3 points]
(1) $\ln(e+1)$
(2) $\ln(e+2)$
(3) $\ln(e+3)$
(4) $\ln(2e+1)$
(5) $\ln(2e+2)$

4. As shown in the figure, on grid paper with unit squares, the three-view of a geometric solid obtained by cutting off part of a cylinder with a plane is shown. The volume of this geometric solid is
A. $90 \pi$
B. $63 \pi$
C. $42 \pi$
D. $36 \pi$ [Figure] [Figure]
5. Let $x , y$ satisfy the constraints $\left\{ \begin{array} { l } 2 x + 3 y - 3 \leqslant 0 , \\ 2 x - 3 y + 3 \geqslant 0 , \\ y + 3 \geqslant 0 , \end{array} \right.$ then the minimum value of $z = 2 x + y$ is
A. $-3$
B. $- 9$
C. $1$
D. $9$

Points $A, B, C, D$ are on the surface of a sphere with radius 4. $\triangle ABC$ is an equilateral triangle with area $9 \sqrt { 3 }$. The maximum volume of the tetrahedron $D$-$ABC$ is
A. $12 \sqrt { 3 }$
B. $18 \sqrt { 3 }$
C. $24 \sqrt { 3 }$
D. $54 \sqrt { 3 }$

A cone has apex $S$, and edges $S A , S B$ are mutually perpendicular. The angle between $S A$ and the base plane is $30 ^ { \circ }$. If the area of $\triangle S A B$ is 8, then the volume of the cone is \_\_\_\_ .

The apex of a cone is $S$. The cosine of the angle between generatrices $S A$ and $S B$ is $\frac { 7 } { 8 }$. The angle between $S A$ and the base of the cone is $45 ^ { \circ }$. The area of $\triangle S A B$ is $5 \sqrt { 15 }$. Then the lateral surface area of the cone is $\_\_\_\_$.

A certain geometric solid is obtained by removing a quadrangular prism from a cube. Its three-view drawing is shown in the figure. If the side length of each small square on the grid paper is 1, then the volume of this geometric solid is $\_\_\_\_$.

Students engage in labor practice at a factory using 3D printing technology to create models. As shown in the figure, the model is a rectangular prism $ABCD - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ with a square pyramid $O - EFGH$ removed, where $O$ is the center of the rectangular prism, and $E , F , G , H$ are the midpoints of the respective edges. $AB = BC = 6 \mathrm{~cm} , AA _ { 1 } = 4 \mathrm{~cm}$. The density of the 3D printing material is $0.9 \mathrm{~g} / \mathrm{cm} ^ { 3 }$. Disregarding printing losses, the mass of material needed to create this model is \_\_\_\_\_\_.

The Great Pyramid of Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. The ratio of the height of a lateral triangular face to the side length of the base square is
A. $\frac { \sqrt { 5 } - 1 } { 4 }$
B. $\frac { \sqrt { 5 } - 1 } { 2 }$
C. $\frac { \sqrt { 5 } + 1 } { 4 }$
D. $\frac { \sqrt { 5 } + 1 } { 2 }$

A cone has a base radius of 1 and slant height of 3. The volume of the largest sphere that can be inscribed in this cone is $\_\_\_\_$ .

127. In an equilateral triangle with side $2\sqrt{3}$ units, the volume of the solid obtained by rotating both shaded regions about the altitude $AH$ is which of the following?
[Figure: Equilateral triangle with altitude AH and shaded regions (inscribed circle area)]
(1) $\dfrac{4\pi}{3}$ (2) $\dfrac{3\pi}{2}$ [6pt] (3) $2\pi$ (4) $\dfrac{5\pi}{3}$
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