Let $R$ be the region bounded by the graph of $y = e ^ { 2 x - x ^ { 2 } }$ and the horizontal line $y = 2$, and let $S$ be the region bounded by the graph of $y = e ^ { 2 x - x ^ { 2 } }$ and the horizontal lines $y = 1$ and $y = 2$, as shown above. (a) Find the area of $R$. (b) Find the area of $S$. (c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = 1$.

Let $R$ be the region bounded by the graphs of $y = \sin ( \pi x )$ and $y = x ^ { 3 } - 4 x$, as shown in the figure above.
(a) Find the area of $R$.
(b) The horizontal line $y = - 2$ splits the region $R$ into two parts. Write, but do not evaluate, an integral expression for the area of the part of $R$ that is below this horizontal line.
(c) The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a square. Find the volume of this solid.
(d) The region $R$ models the surface of a small pond. At all points in $R$ at a distance $x$ from the $y$-axis, the depth of the water is given by $h ( x ) = 3 - x$. Find the volume of water in the pond.

Let $R$ be the region in the first quadrant bounded by the $x$-axis and the graphs of $y = \ln x$ and $y = 5 - x$, as shown in the figure above.
(a) Find the area of $R$.
(b) Region $R$ is the base of a solid. For the solid, each cross section perpendicular to the $x$-axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.
(c) The horizontal line $y = k$ divides $R$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $k$.

The functions $f$ and $g$ are given by $f ( x ) = \sqrt { x }$ and $g ( x ) = 6 - x$. Let $R$ be the region bounded by the $x$-axis and the graphs of $f$ and $g$, as shown in the figure above.
(a) Find the area of $R$.
(b) The region $R$ is the base of a solid. For each $y$, where $0 \leq y \leq 2$, the cross section of the solid taken perpendicular to the $y$-axis is a rectangle whose base lies in $R$ and whose height is $2 y$. Write, but do not evaluate, an integral expression that gives the volume of the solid.
(c) There is a point $P$ on the graph of $f$ at which the line tangent to the graph of $f$ is perpendicular to the graph of $g$. Find the coordinates of point $P$.

Let $R$ be the shaded region bounded by the graph of $y = x e ^ { x ^ { 2 } }$, the line $y = - 2 x$, and the vertical line $x = 1$, as shown in the figure above.
(a) Find the area of $R$.
(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is rotated about the horizontal line $y = - 2$.
(c) Write, but do not evaluate, an expression involving one or more integrals that gives the perimeter of $R$.

The graphs of the functions $f$ and $g$ are shown in the figure for $0 \leq x \leq 3$. It is known that $g(x) = \frac{12}{3 + x}$ for $x \geq 0$. The twice-differentiable function $f$, which is not explicitly given, satisfies $f(3) = 2$ and $\int_{0}^{3} f(x)\, dx = 10$.
(a) Find the area of the shaded region enclosed by the graphs of $f$ and $g$.
(b) Evaluate the improper integral $\int_{0}^{\infty} (g(x))^{2}\, dx$, or show that the integral diverges.
(c) Let $h$ be the function defined by $h(x) = x \cdot f'(x)$. Find the value of $\int_{0}^{3} h(x)\, dx$.

An advertiser wishes to print a logo on a T-shirt. He draws this logo using the curves of two functions $f$ and $g$ defined on $\mathbb{R}$ by: $$f(x) = \mathrm{e}^{-x}(-\cos x + \sin x + 1) \text{ and } g(x) = -\mathrm{e}^{-x}\cos x$$ It is admitted that the functions $f$ and $g$ are differentiable on $\mathbb{R}$.

Part A — Study of function $f$

1. Justify that, for all $x \in \mathbb{R}$: $$-\mathrm{e}^{-x} \leqslant f(x) \leqslant 3\mathrm{e}^{-x}$$
2. Deduce the limit of $f$ as $x \to +\infty$.
3. Prove that, for all $x \in \mathbb{R}$, $f'(x) = \mathrm{e}^{-x}(2\cos x - 1)$ where $f'$ is the derivative of $f$.
4. In this question, we study function $f$ on the interval $[-\pi; \pi]$. a. Determine the sign of $f'(x)$ for $x$ in the interval $[-\pi; \pi]$. b. Deduce the variations of $f$ on $[-\pi; \pi]$.

Part B — Area of the logo

We denote by $\mathscr{C}_f$ and $\mathscr{C}_g$ the graphs of functions $f$ and $g$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. The graphical unit is 2 centimetres.

1. Study the relative position of curve $\mathscr{C}_f$ with respect to curve $\mathscr{C}_g$ on $\mathbb{R}$.
2. Let $H$ be the function defined on $\mathbb{R}$ by: $$H(x) = \left(-\frac{\cos x}{2} - \frac{\sin x}{2} - 1\right)\mathrm{e}^{-x}$$ It is admitted that $H$ is an antiderivative of the function $x \mapsto (\sin x + 1)\mathrm{e}^{-x}$ on $\mathbb{R}$. We denote by $\mathscr{D}$ the region bounded by curve $\mathscr{C}_f$, curve $\mathscr{C}_g$ and the lines with equations $x = -\frac{\pi}{2}$ and $x = \frac{3\pi}{2}$. a. Shade the region $\mathscr{D}$ on the graph in the appendix to be returned with your work. b. Calculate, in square units, the area of region $\mathscr{D}$, then give an approximate value to $10^{-2}$ in $\mathrm{cm}^2$.

The derivative $f'(x)$ of a function $f(x)$ that is differentiable on the set of all real numbers is $$f'(x) = -x + e^{1 - x^{2}}$$ For a positive number $t$, let $g(t)$ be the area of the region enclosed by the tangent line to the curve $y = f(x)$ at the point $(t, f(t))$, the curve $y = f(x)$, and the $y$-axis. What is the value of $g(1) + g'(1)$? [4 points]
(1) $\frac{1}{2}e + \frac{1}{2}$
(2) $\frac{1}{2}e + \frac{2}{3}$
(3) $\frac{1}{2}e + \frac{5}{6}$
(4) $\frac{2}{3}e + \frac{1}{2}$
(5) $\frac{2}{3}e + \frac{2}{3}$

Given the functions
$$f ( x ) = 2 + 2 x - 2 x ^ { 2 } \text { and } g ( x ) = 2 - 6 x + 4 x ^ { 2 } + 2 x ^ { 3 } ,$$
find:\ a) (1 point) Study the differentiability of $h ( x ) = | f ( x ) |$.\ b) (1.5 points) Find the area of the region bounded by the curves\ $y = f ( x ) , y = g ( x ) , x = 0$ and $x = 2$.

On the coordinate plane, let $\Gamma$ denote the graph of the polynomial function $y = x ^ { 3 } - 4 x ^ { 2 } + 5 x$ , and let $L$ denote the line $y = m x$ , where $m$ is a real number. Based on the above, answer the following questions.
(1) When $m = 2$ , find the $x$-coordinates of the three distinct intersection points of $\Gamma$ and $L$ in the range $x \geq 0$ . (2 points)
(2) Based on (1), find the area of the bounded region enclosed by $\Gamma$ and $L$ . (4 points)
(3) In the range $x \geq 0$ , if $\Gamma$ and $L$ have three distinct intersection points, then the maximum range of $m$ satisfying this condition is $a < m < b$ . Find the values of $a$ and $b$ . (6 points)