Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = \ln x$.
(a) Find the area of the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$.
(b) Find the volume of the solid generated when the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$ is revolved about the line $y = 4$.
(c) Let $h$ be the function given by $h ( x ) = f ( x ) - g ( x )$. Find the absolute minimum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$, and find the absolute maximum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$. Show the analysis that leads to your answers.

Let $f$ and $g$ be the functions given by $f ( x ) = \frac { 1 } { 4 } + \sin ( \pi x )$ and $g ( x ) = 4 ^ { - x }$. Let $R$ be the shaded region in the first quadrant enclosed by the $y$-axis and the graphs of $f$ and $g$, and let $S$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$, as shown in the figure above.
(a) Find the area of $R$.
(b) Find the area of $S$.
(c) Find the volume of the solid generated when $S$ is revolved about the horizontal line $y = - 1$.

Let $R$ be the region in the first quadrant enclosed by the graphs of $f(x) = 8x^3$ and $g(x) = \sin(\pi x)$, as shown in the figure.
(a) Write an equation for the line tangent to the graph of $f$ at $x = \frac{1}{2}$.
(b) Find the area of $R$.
(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated about the horizontal line $y = 1$.

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 $2y$. 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 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 $k$.

Let $f ( x ) = 2 x ^ { 2 } - 6 x + 4$ and $g ( x ) = 4 \cos \left( \frac { 1 } { 4 } \pi x \right)$. Let $R$ be the region bounded by the graphs of $f$ and $g$, 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 $y = 4$.
(c) The region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$-axis is a square. Write, but do not evaluate, an integral expression that gives the volume of the solid.

Let $f$ and $g$ be the functions defined by $f(x) = 1 + x + e^{x^2 - 2x}$ and $g(x) = x^4 - 6.5x^2 + 6x + 2$. Let $R$ and $S$ be the two regions enclosed by the graphs of $f$ and $g$ shown in the figure above.
(a) Find the sum of the areas of regions $R$ and $S$.
(b) Region $S$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. Find the volume of the solid.
(c) Let $h$ be the vertical distance between the graphs of $f$ and $g$ in region $S$. Find the rate at which $h$ changes with respect to $x$ when $x = 1.8$.

Let $R$ be the region enclosed by the graphs of $g(x) = -2 + 3\cos\left(\dfrac{\pi}{2}x\right)$ and $h(x) = 6 - 2(x-1)^2$, the $y$-axis, and the vertical line $x = 2$, as shown in the figure above.
(a) Find the area of $R$.
(b) Region $R$ is the base of a solid. For the solid, at each $x$ the cross section perpendicular to the $x$-axis has area $A(x) = \dfrac{1}{x+3}$. Find the volume of the solid.
(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 $y = 6$.

Let $f$ and $g$ be the functions defined by $f(x) = \ln(x+3)$ and $g(x) = x^4 + 2x^3$. The graphs of $f$ and $g$ intersect at $x = -2$ and $x = B$, where $B > 0$.
(a) Find the area of the region enclosed by the graphs of $f$ and $g$.
(b) For $-2 \leq x \leq B$, let $h(x)$ be the vertical distance between the graphs of $f$ and $g$. Is $h$ increasing or decreasing at $x = -0.5$? Give a reason for your answer.
(c) The region enclosed by the graphs of $f$ and $g$ is the base of a solid. Cross sections of the solid taken perpendicular to the $x$-axis are squares. Find the volume of the solid.
(d) A vertical line in the $xy$-plane travels from left to right along the base of the solid described in part (c). The vertical line is moving at a constant rate of 7 units per second. Find the rate of change of the area of the cross section above the vertical line with respect to time when the vertical line is at position $x = -0.5$.

The functions $f$ and $g$ are defined by $f(x) = x^2 + 2$ and $g(x) = x^2 - 2x$, as shown in the graph.
(a) Let $R$ be the region bounded by the graphs of $f$ and $g$, from $x = 0$ to $x = 2$, as shown in the graph. Write, but do not evaluate, an integral expression that gives the area of region $R$.
(b) Let $S$ be the region bounded by the graph of $g$ and the $x$-axis, from $x = 2$ to $x = 5$, as shown in the graph. Region $S$ is the base of a solid. For this solid, at each $x$ the cross section perpendicular to the $x$-axis is a rectangle with height equal to half its base in region $S$. Find the volume of the solid. Show the work that leads to your answer.
(c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when region $S$, as described in part (b), is rotated about the horizontal line $y = 20$.

The shaded region $R$ is bounded by the graphs of the functions $f$ and $g$, where $f ( x ) = x ^ { 2 } - 2 x$ and $g ( x ) = x + \sin ( \pi x )$, as shown in the figure.
(Note: Your calculator should be in radian mode.)
A. Find the area of $R$. Show the setup for your calculations.
B. Region $R$ is the base of a solid. For this solid, at each $x$ the cross section perpendicular to the $x$-axis is a rectangle with height $x$ and base in region $R$. Find the volume of the solid. Show the setup for your calculations.
C. Write, but do not evaluate, an integral expression for the volume of the solid generated when the region $R$ is rotated about the horizontal line $y = - 2$.
D. It can be shown that $g ^ { \prime } ( x ) = 1 + \pi \cos ( \pi x )$. Find the value of $x$, for $0 < x < 1$, at which the line tangent to the graph of $f$ is parallel to the line tangent to the graph of $g$.

Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = \ln x$.
(a) Find the area of the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$.
(b) Find the volume of the solid generated when the region enclosed by the graphs of $f$ and $g$ between $x = \frac { 1 } { 2 }$ and $x = 1$ is revolved about the line $y = 4$.
(c) Let $h$ be the function given by $h ( x ) = f ( x ) - g ( x )$. Find the absolute minimum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$, and find the absolute maximum value of $h ( x )$ on the closed interval $\frac { 1 } { 2 } \leq x \leq 1$. Show the analysis that leads to your answers.

Let $R$ be the region in the first quadrant bounded by the $y$-axis and the graphs of $y = 4x - x^3 + 1$ and $y = \frac{3}{4}x$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.
(c) Write an expression involving one or more integrals that gives the perimeter of $R$. Do not evaluate.

Let $f$ be the function given by $f(x) = 4x^2 - x^3$, and let $\ell$ be the line $y = 18 - 3x$, where $\ell$ is tangent to the graph of $f$. Let $R$ be the region bounded by the graph of $f$ and the $x$-axis, and let $S$ be the region bounded by the graph of $f$, the line $\ell$, and the $x$-axis.
(a) Show that $\ell$ is tangent to the graph of $y = f(x)$ at the point $x = 3$.
(b) Find the area of $S$.
(c) Find the volume of the solid generated when $R$ is revolved about the $x$-axis.

Let $f$ and $g$ be the functions given by $f ( x ) = \frac { 1 } { 4 } + \sin ( \pi x )$ and $g ( x ) = 4 ^ { - x }$. Let $R$ be the shaded region in the first quadrant enclosed by the $y$-axis and the graphs of $f$ and $g$, and let $S$ be the shaded region in the first quadrant enclosed by the graphs of $f$ and $g$, as shown in the figure above.
(a) Find the area of $R$.
(b) Find the area of $S$.
(c) Find the volume of the solid generated when $S$ is revolved about the horizontal line $y = - 1$.

Let $R$ be the shaded region bounded by the graph of $y = \ln x$ and the line $y = x - 2$.
(a) Find the area of $R$.
(b) Find the volume of the solid generated when $R$ is rotated about the horizontal line $y = -3$.
(c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when $R$ is rotated about the $y$-axis.

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 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) $\frac{2}{3}e + \frac{2}{3}$

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)