Math topic中文

Volume of revolution

Volume of revolution about the $x$ - or $y$ -axis.

Practice questions

1The curve $C$ is defined by the parametric equations $x = t - \sin t$ , $y = \sin^{2} t$ , $\;0 \le t \le \pi$ . The point…
2The diagram shows the graph of $y = f(x)$ . The curve has a turning point at $(0,a)$ and the lines $y = \pm kx$ are asymp…
3You are given that $C = \int_{0}^{\pi} \mathrm{e}^{-2x} \cos x \; \mathrm{d}x$ and $S = \int_{0}^{\pi} \mathrm{e}^{-2x} \sin x \; \mathrm{d}x$ …
4A function f is defined as $f(x)=\ln(5-x),\; x<5.$ Find the volume of revolution when the region bounded by the curve of…
5By using the substitution $u = \sqrt{x+1}$ , find $\displaystyle \int \frac{\sqrt{x+1}}{x-1}\,dx$ . The region $R$ is boun…
6Sketch the curve with equation $\frac{x^2}{400} + \frac{(y-10)^2}{900} = 1.$ [There is no need to indicate the x-inter…
7Find $\displaystyle\int x^{2}e^{-2x}\,\mathrm{d}x$ . The curve with equation $y = xe^{-x}$ and the line with equation $y=\tfrac{1}{e}x$ …
8The curve $C$ has equation $y=\ln\left(2-\lvert x\rvert\right)$ . The region $R$ is bounded by $C$ , the lines $y=\ln 2$ ,…
9A curve $C$ has equation $y = 2ax + \frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x \ne 0$ . Using diff…
10The 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…
11Find $\int x^{2} e^{-2x}\, dx$ . The curve with equation $y = x e^{-x}$ and the line with equation $y = \tfrac{1}{e} x$ …
12The curve $C$ has equation $y = x\cos 2x$ , where $0 \le x \le \pi$ . Find the exact $x$ -coordinates of the points where $C$ …
13Find the exact value of $\pi \int_{1}^{6} x^3 \text{ d}x$ . The diagram shows a sketch of the curve $y = x^{\frac{3}{2}}$ …
14Find $\displaystyle \int \dfrac{6+2x}{\sqrt{1-4x-x^2}}\,\mathrm{d}x$ . The diagram above shows the region $R$ bounded by…
15Find $\displaystyle \int x \sin^{2} x \, dx$ . The region R is bounded by the curve $y = \sqrt{x \sin x}$ , the lines x =…
16The region $R$ is bounded by the lines $y=0$ , $y=\dfrac{5}{2}\sqrt{3}$ and the curve $x^2 = 25 - y^2$ . $R$ is rotated ab…
17Find 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…
18Integrate 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…
19On the same axes, sketch the graphs of $y = \sin 2x$ and $y = \cos x$ for $0 \le x \le \frac{\pi}{2}$ , stating the exact…
20The 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…
21Curve C is a circle with radius 2 and center at the origin with equation $x^2 + y^2 = 4$ . Line L has the equation $\sqrt{3}y = x + 2$ …
22The diagram shows the curve $C$ with equation $y = 2\sin^{-1}x$ and the line $L$ with equation $y = \frac{8\pi}{3}x - \pi$ …
23Use the substitution $x=1+\sin u$ to find $\int\sqrt{2x-x^{2}}\,dx$ . The finite region $R$ , which includes the point (0…
24The curves $C_1$ and $C_2$ have equations $y=\dfrac{x-b}{x-a}$ and $y=\dfrac{x-b}{b}$ respectively, where $a$ and $b$ ar…
25A curve $G$ has equation $y = ax + b + \dfrac{c}{(x+1)^2}$ , where $a$ , $b$ and $c$ are positive integers. Let $R$ be the…
26The diagram above shows a sketch of the graph of $y = \text{f}(x)$ for $x \ge 0$ . There are $n$ rectangles each of width…
27The shaded region R bounded by the curve $x = 5 - (y - 1)^2$ , the line $y = -x$ and the x-axis is rotated about the x-ax…
28The diagram shows the curve $C$ with equation $y = \frac{12}{(3x+2)^2 + 4}$ which has a turning point at $\left(-\frac{2}{3}, 3\right)$ …
29Let $R$ 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…
30The 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…
31Nozzles are often used at the end of hoses to direct and speed up the flow of water. The diagram below represents the si…
32A curve $G$ has equation $y = \frac{1}{\cos x} + \cos x$ . The region bounded by $G$ , the line $x = \frac{\pi}{3}$ , the $x$ …
33The graph shows the curve $C$ with equation $y = x^2$ and the line $L$ with equation $y = 2x-1$ . The line $L$ is tangent…
34A 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…
35Sketch the curve with equation $\displaystyle\frac{x^2}{400}+\frac{(y-10)^2}{900}=1.$ [There is no need to indicate th…
36[In this question, all measurements of length are in metres.] A monument is to be constructed which comprises two parts…
37The curve $C$ has equation $y=\ln\left(2-\lvert x\rvert\right)$ . The region $R$ is bounded by $C$ , the lines $y=\ln 2$ ,…
38Curve $C$ is a circle with radius 2 and center at the origin with equation $x^2 + y^2 = 4$ . Line $L$ has the equation $\sqrt{3}y = x + 2$ …
39A curve $C$ has parametric equations $x = 2\cos t + 1, \quad y = \sin t , \quad 0 \le t < 2\pi .$ Sketch $C$ , indicati…
40A function f is defined as $f(x)=\ln(5-x)$ , $x<5$ . Find the volume of revolution when the region bounded by the curve of…
41A function f is defined as $f(x) = \ln(5-x)$ , $x < 5$ . Find the volume of revolution when the region bounded by the curv…
42The region $R$ is bounded by the curve with equation $y = x \ln x$ , the lines $x = \text{e}$ , $x = \text{e}^2$ and the $x$ …
43Find $\displaystyle \int \frac{9u-8}{4+9u^2}\,\mathrm{d}u$ . The curve $C$ is given by the parametric equations $x = u^2 + u + 1$ …
44The curve $C$ (as shown in the diagram below) has equation $y = x^2 \sin x$ , $-\pi \le x \le \pi$ . Calculate the exact a…