Math topic中文

Area of region

Area of a region bounded by a curve and lines parallel to the coordinate axes, between a curve and a line, or between two curves.

Practice questions

1[Cambridge/9709/2010] The diagram shows parts of the curves $y = 9 - x^3$ and $y = \frac{8}{x^3}$ and their points o…
2Given that the first three terms in the expansion of $\dfrac{1}{a+bx}$ are $\dfrac{1}{2}+\dfrac{1}{4}x+cx^2$ , find the v…
3The parametric equations of a curve C are $x = at,\; y = at^{3}$ , where $a$ is a positive constant. The point P on the c…
4A curve with equation $y=f(x)$ is transformed by a reflection in the y-axis, followed by a translation of 1 unit in the…
5The curve $C$ has equation $y = \frac{x - 2}{kx^2 + x - 2}$ , where $k > 1$ . Find the equation of the tangent at the poin…
6Show that for $x>1$ , $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1}{\sqrt{x^2-1}}\right)=-\dfrac{x}{(x^2-1)^{3/2}}\,.$ …
7The curve $C$ is defined by the parametric equations $x = t - \sin t$ , $y = \sin^{2} t$ , $\;0 \le t \le \pi$ . The point…
8A curve C is defined by the parametric equations $x = \theta^2 - 1,\quad y = \tan\frac{\theta}{2},\quad 0 \le \theta < \pi$ …
9By using the substitution $u = \sqrt{x+1}$ , find $\displaystyle \int \frac{\sqrt{x+1}}{x-1}\,dx$ . The region $R$ is boun…
10A curve $C$ is defined parametrically by $x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} < t < \frac{\pi}{4},$ …
11The curve $C$ has equation $y=\ln\left(2-\lvert x\rvert\right)$ . The region $R$ is bounded by $C$ , the lines $y=\ln 2$ ,…
12A farmer has a plot of the land, $OAB$ , with adjacent sides $OA$ and $OB$ . With reference to $O$ as the origin, $OA$ and…
13The diagram above shows a sketch of the graph of $y=f(x)$ for $x \\ge 0$ . There are $n$ rectangles each of width $h$ dra…
14The function $\text{f}$ is defined by $\text{f} : x \to 2x - \frac{1}{2x}, \quad 0 < x < 2.$ It is given that $\text{f}^{-1}$ …
15Find the area of the region bounded by the graphs of $y = 2x^2 + 3$ and $y = -2x + 7$ . State the area of the region boun…
16Integrate by parts twice to show that $\int e^{2x}\sin x\,dx=\tfrac{1}{5}e^{2x}(2\sin x-\cos x)+c.$ The diagram below…
17The curve $C$ has parametric equations $x = \sin \theta + 1 \quad \text{and} \quad y = \sqrt{3} \cos \theta - 1, \text{ where } -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}.$ …
18The diagram below shows a region $R$ bounded by the curve $(y+5)^{2}=x-3$ and the line $y=x-10$ . Find the volume of soli…
19A curve $C$ is defined by the parametric equations $x = 2\cos\theta + 1, \quad y = \sin\theta, \quad \text{where } -\pi \le \theta \le \pi.$ …
20Curve C is a circle with radius 2 and center at the origin with equation $x^2 + y^2 = 4$ . Line L has the equation $\sqrt{3}y = x + 2$ …
21Use the substitution $x=1+\sin u$ to find $\int\sqrt{2x-x^{2}}\,dx$ . The finite region $R$ , which includes the point (0…
22By using the substitution $x = \frac{1}{u}$ , show that $\int_{2\sqrt{2}}^{4} \frac{1}{x\sqrt{x^2 - 4}} \, \mathrm{d}x = \frac{\pi}{24}$ …
23A curve $C$ is defined parametrically by $x = a\tan t, \qquad y = a\sec^{2}t\sin t, \qquad -\frac{\pi}{4} < t < \frac{\pi}{4},$ …
24A curve $G$ has equation $y = ax + b + \dfrac{c}{(x+1)^2}$ , where $a$ , $b$ and $c$ are positive integers. Let $R$ be the…
25The diagram above shows a sketch of the graph of $y = \text{f}(x)$ for $x \ge 0$ . There are $n$ rectangles each of width…
26The diagram shows the curve $C$ with equation $y = \frac{12}{(3x+2)^2 + 4}$ which has a turning point at $\left(-\frac{2}{3}, 3\right)$ …
27A curve $C$ is defined by the parametric equations $\begin{aligned} x &= 4\sin t, \\ y &= b\cos t - 1, \end{aligned}$ …
28Let $R$ be the region bounded by the curves $y = 4\sqrt{x}$ and $y = x^2 + 1$ . Find the area of region $R$ . If the line…
29The diagram above shows a sketch of the graph of $y=f(x)$ for $x\ge 0$ . There are $n$ rectangles each of width $h$ drawn…
30Nozzles are often used at the end of hoses to direct and speed up the flow of water. The diagram below represents the si…
31A curve C has parametric equations $x = e^{t},\ \ y = \sin t$ where $0 \le t \le \pi$ . Find $\int e^{t}\sin t\,dt$ . Ske…
32A curve P is defined by $y = -\sin^2 x$ where $-\pi \le x \le 0$ . The finite region R is bounded by the curve P, the y-a…
33[In this question, all measurements of length are in metres.] A monument is to be constructed which comprises two parts…
34The diagram above shows the curve $C_1$ with equation $y = \frac{\ln x}{x^2}$ , where $x \ge 1$ . Show that the exact coor…
35A curve $C$ is defined by the parametric equations $x = \theta^2 - 1$ , $y = \tan\frac{\theta}{2}$ , $0 \le \theta < \pi$ .…
36The curve $C$ has equation $y=\ln\left(2-\lvert x\rvert\right)$ . The region $R$ is bounded by $C$ , the lines $y=\ln 2$ ,…
37Curve $C$ is a circle with radius 2 and center at the origin with equation $x^2 + y^2 = 4$ . Line $L$ has the equation $\sqrt{3}y = x + 2$ …
38The curve $C$ is defined by the parametric equations $x = 1 - \cos t, \quad y = \sin 2t, \quad \text{where } 0 \le t \le \frac{\pi}{2}.$ …
39The function $f$ is defined by $f: x \mapsto 2x - \dfrac{1}{2x},\quad 0 < x < 2$ . It is given that $f^{-1}$ exists. Defi…
40The diagram shows the curve $C$ with parametric equations $x = (1+t)^2, \quad y = (3-t)^2.$ The curve $C$ meets the ax…
41A curve $C$ has parametric equations $x = t^2, \quad y = t^3 + 1 \quad \text{for } t \in \mathbb{R}.$ Sketch the graph…
42Find $\displaystyle \int \frac{9u-8}{4+9u^2}\,\mathrm{d}u$ . The curve $C$ is given by the parametric equations $x = u^2 + u + 1$ …
43The parametric equations of a curve C are $x = 3\sec t$ and $y = -2\tan t$ , where $0 < t < \frac{\pi}{2}$ . Find the cart…
44A curve $C$ has parametric equations $x = 3\cos\theta - 2\cos 3\theta ,$ $y = 9\sin\theta - \sin 3\theta ,$ for $0 \le \theta \le 2\pi$ …
45The curve $C$ (as shown in the diagram below) has equation $y = x^2 \sin x$ , $-\pi \le x \le \pi$ . Calculate the exact a…
46A curve $C$ is defined parametrically by $x = a \tan t, \quad y = a \sec^2 t \sin t, \quad -\frac{\pi}{4} < t < \frac{\pi}{4},$ …