LFM Pure and Mechanics

ap-calculus-ab 2002 Q1 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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.

ap-calculus-ab 2004 Q6 Maximize or Optimize Area View

Let $\ell$ be the line tangent to the graph of $y = x^{n}$ at the point $(1,1)$, where $n > 1$.
(a) Find $\displaystyle\int_{0}^{1} x^{n}\,dx$ in terms of $n$.
(b) Let $T$ be the triangular region bounded by $\ell$, the $x$-axis, and the line $x = 1$. Show that the area of $T$ is $\dfrac{1}{2n}$.
(c) Let $S$ be the region bounded by the graph of $y = x^{n}$, the line $\ell$, and the $x$-axis. Express the area of $S$ in terms of $n$ and determine the value of $n$ that maximizes the area of $S$.

ap-calculus-ab 2005 Q1 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $y = - 1$.

ap-calculus-ab 2011 Q3 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $y = 1$.

ap-calculus-ab 2011 Q3 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $P$.

ap-calculus-ab 2012 QFR2 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $k$.

ap-calculus-ab 2013 Q5 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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.

ap-calculus-ab 2015 Q2 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $x = 1.8$.

ap-calculus-ab 2019 Q5 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $y = 6$.

ap-calculus-ab 2022 Q2 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $x = -0.5$.

ap-calculus-ab 2024 Q6 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $y = 20$.

ap-calculus-ab 2025 Q2 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $g$.

ap-calculus-bc 2002 Q1 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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.

ap-calculus-bc 2002 Q3 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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.

ap-calculus-bc 2003 Q1 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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.

ap-calculus-bc 2004 Q6 Maximize or Optimize Area View

Let $\ell$ be the line tangent to the graph of $y = x ^ { n }$ at the point $( 1,1 )$, where $n > 1$, as shown above.
(a) Find $\int _ { 0 } ^ { 1 } x ^ { n } d x$ in terms of $n$.
(b) Let $T$ be the triangular region bounded by $\ell$, the $x$-axis, and the line $x = 1$. Show that the area of $T$ is $\frac { 1 } { 2 n }$.
(c) Let $S$ be the region bounded by the graph of $y = x ^ { n }$, the line $\ell$, and the $x$-axis. Express the area of $S$ in terms of $n$ and determine the value of $n$ that maximizes the area of $S$.

ap-calculus-bc 2005 Q1 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $y = - 1$.

ap-calculus-bc 2006 Q1 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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.

ap-calculus-bc 2008 Q1 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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.

ap-calculus-bc 2014 Q5 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $R$.

ap-calculus-bc 2023 Q5 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $\int_{0}^{3} h(x)\, dx$.

bac-s-maths 2015 Q1C Compute Area Directly (Numerical Answer) View

Determine the area $\mathscr{A}$, expressed in square units, of the shaded region in the graph of Part A (the region bounded by the curve $\mathscr{C}_u$ where $u(x) = \frac{x^2 - 5x + 4}{x^2}$ between its zeros $x=1$ and $x=4$).
For all real $\lambda$ greater than or equal to 4, we denote by $\mathscr{A}_{\lambda}$ the area, expressed in square units, of the region formed by the points $M$ with coordinates $(x; y)$ such that $$4 \leqslant x \leqslant \lambda \quad \text{and} \quad 0 \leqslant y \leqslant u(x).$$ Does there exist a value of $\lambda$ for which $\mathscr{A}_{\lambda} = \mathscr{A}$?

bac-s-maths 2018 Q3 Multi-Part Free Response with Area, Volume, and Additional Calculus View

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 $f$

Justify that, for all $x \in \mathbb{R}$: $$-\mathrm{e}^{-x} \leqslant f(x) \leqslant 3\mathrm{e}^{-x}$$
Deduce the limit of $f$ as $x \to +\infty$.
Prove that, for all $x \in \mathbb{R}$, $f'(x) = \mathrm{e}^{-x}(2\cos x - 1)$ where $f'$ is the derivative of $f$.
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.

Study the relative position of curve $\mathscr{C}_f$ with respect to curve $\mathscr{C}_g$ on $\mathbb{R}$.
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 $\mathrm{cm}^2$.

bac-s-maths 2025 Q2 6 marks Prove or Verify an Area Result View

In the orthonormal coordinate system (O; I, J), we have represented:

the line with equation $y = x$;
the line with equation $y = 1$;
the line with equation $x = 1$;
the parabola with equation $y = x ^ { 2 }$.

We can thus divide the square OIKJ into three zones.
Part A Prove the results shown in the table below.

ZONE	ZONE 1	ZONE 2	ZONE 3
AREA	$\frac { 1 } { 2 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 6 }$

Part B: a first game
A player throws a dart at the square above. It is admitted that the probability that it lands on a zone is equal to the area of that zone. Thus, the probability that the dart lands on ZONE 3 is equal to $\frac { 1 } { 6 }$.

If the dart lands on ZONE 3, then the player tosses a fair coin. If the coin lands on HEADS, then the player wins, otherwise he loses.
If the dart lands on a zone other than ZONE 3, then the player rolls a fair six-sided die. If the die lands on FACE 6, then the player wins, otherwise he loses.

We note the following events: $T$: ``the dart lands on ZONE 3''; $G$: ``the player wins''.

Represent the situation with a weighted tree.
Prove that the probability of event $G$ is equal to $\frac { 2 } { 9 }$.
Given that the player has won, what is the probability that the dart landed on ZONE 3?

Part C: a second game
A player, called player $n^{\mathrm{o}}1$, throws a dart at the previous square. As in Part B, it is admitted that the probability that the dart lands on each of the zones is equal to the area of that zone. The player wins a sum equal, in euros, to the number of the zone. For example, if the dart lands on ZONE 3, the player wins 3 euros. We denote $X _ { 1 }$ the random variable giving the winnings of player $n^{\circ}1$. We denote respectively $E \left( X _ { 1 } \right)$ and $V \left( X _ { 1 } \right)$ the expectation and variance of the random variable $X _ { 1 }$.

a. Calculate $E \left( X _ { 1 } \right)$. b. Show that $V \left( X _ { 1 } \right) = \frac { 5 } { 9 }$.
A player $n^{\circ}2$ and a player $n^{\circ}3$ play in turn, under the same conditions as player $n^{\circ}1$. It is admitted that the games of these three players are independent of each other. We denote $X _ { 2 }$ and $X _ { 3 }$ the random variables giving the winnings of players $n^{\circ}2$ and $n^{\circ}3$. We denote $Y$ the random variable defined by $Y = X _ { 1 } + X _ { 2 } + X _ { 3 }$. a. Determine the probability that $Y = 9$. b. Calculate $E ( Y )$. c. Justify that $V ( Y ) = \frac { 5 } { 3 }$.

brazil-enem 2011 Q178 Compute Area Directly (Numerical Answer) View

A área da região delimitada pela parábola $y = x^2$ e pela reta $y = 4$ é
(A) $\dfrac{16}{3}$ (B) $\dfrac{32}{3}$ (C) $8$ (D) $\dfrac{64}{3}$ (E) $16$

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