Math topic中文

Area curve and lines

finding the area of a region bounded by a curve and line(s) (excluding area of region between 2 curves)

Practice questions

1The figure shows part of the curve $y = g(x)$ . $(2, 8)$ and $(6, 1)$ are two points on the curve. Given that $\int_{2}^{6} y \text{ } dx = 32$ …
2The diagram shows part of the curve $y = 3e^{-\frac{1}{2}x} - 6$ which crosses the $x$ -axis at $P$ and the $y$ -axis at $Q$ …
3The diagram shows part of the curve $y = \sqrt{3x + 4}$ . The line $AB$ is tangent to the curve at $A$ . The points $B$ an…
4The diagram shows part of the curve $y = (1 - x)(x - 4)$ and two lines $l_1$ and $l_2$ . The lines $l_1$ and $l_2$ are ta…
5The diagram below shows the graph of $y = \frac{1}{2x+1}$ and the tangent to the curve where the curve intersects the $y$ …
6The diagram shows part of the curve $y = \frac{1}{\left(\frac{3}{4}x - 5\right)^2} + 1$ . $P$ is the point on the curve w…
7The diagram shows part of the curve $y = 4 - e^{-2x}$ which crosses the axes at $A$ and at $B$ . Find the coordinates of…
8The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
9Find the area of the shaded region. The figure above is not drawn to scale.
10The graph shows part of the curve $y = -3(2x-4)^{-1/3}$ . The point $P(-2,\ 1.5)$ lies on the curve and the normal to the…
11The diagram shows part of the curve $y = \sqrt{2x + 4}$ , meeting the $x$ -axis at the point $A$ and the $y$ -axis at the p…
12The diagram shows part of the curve $y = 2 \sin 2x + 1$ . $P$ is the maximum point of the curve and $Q$ is the point on t…
13The diagram shows part of the curve $y = x^3 + 3x^2 - 2$ . The straight line $L$ cuts the curve at $A(-3, -2)$ , the x-axi…
14The diagram shows part of the curve $y = 2\sin\frac{x}{2}$ , where $A$ lies on the curve. The $x$ -coordinate of $A$ is $\frac{4\pi}{3}$ …
15求曲线 $y = x^2$ 与 $y = 2x$ 围成的封闭图形的面积。
16The diagram shows the curve $y = \cos x + \sin x$ for $0 \le x \le \pi$ . The curve meets the $y$ -axis at the point $A$ a…
17The diagram shows part of the curve $y = \frac{15}{(0.5x+1)}$ and a straight line $y = 1.25x - 10$ . The straight line in…
18Find $\dfrac{\mathrm{d}}{\mathrm{d}x}\left(3x^{2}\ln x\right). Hence find$ \int x\ln x,\mathrm{d}x $. The diagram shows the line$ …
19The diagram shows part of the curve $y = \frac{5}{2x - 1}$ . The line $9y = -10x + 55$ intersects the curve at points $A$ …
20The diagram shows part of the curve $y = \frac{7}{6x + 1}$ intersecting the $y$ -axis at $A(0, 7)$ . The tangent to the cu…
21The graph shows the curve $C$ with equation $y = x^2$ and the line $L$ with equation $y = 2x-1$ . The line $L$ is tangent…
22The diagram shows part of the curve $y = \sqrt{3x-2}$ . The normal to the curve at $B$ meets the $x$ -axis at $C$ . Given t…
23The diagram shows the curve $y = 4 - 2\cos 2x$ for $0 \le x \le \frac{3}{2}\pi$ radians and a straight line passing thro…
24The diagram shows part of the curve $y = \sqrt{2x+5}$ . A is a point on the curve and the x-coordinate of A is 2. The nor…
25The diagram shows part of the curve $y = 5 + \frac{4}{e^{2x}}$ intersecting the $y$ -axis at point $P$ . The tangent to th…
26The diagram shows part of the curve of $y = x(x-2)^2$ , which passes through the point $P(1, 1)$ . Show that the equation…
27The diagram shows part of the curve $y = \dfrac{3}{9-2x}$ . The point $P\ (3,\ 1)$ lies on the curve and the normal to th…
28The diagram shows part of the curve $y = \frac{16}{(x+1)^2} - 1$ , cutting the $x$ -axis at $M$ . The tangent at the point…
29The diagram shows part of the curve $y = -\sqrt{10 - 6x}$ meeting the x-axis at the point A. The normal to the curve at…
30The diagram shows part of the curve $y = 16 + 6x - x^2$ intersecting the line $y = 26 - x$ at $A$ and $B$ . Find the coor…
31The diagram shows part of the curve $y = 1 - \sin x$ passing through the point $P$ . The curve touches the $x$ -axis at th…
32The diagram shows part of the curve $y = e^{\left(-\frac{1}{2}x + 2\right)} + x - 1$ that intersects the $y$ -axis at $R$ …
33The diagram shows the curve $y = \sin 2x + x$ for $0 \le x \le \pi$ radians. The point $M$ is the minimum point of the c…
34The diagram below shows the curve $y = x^3 - 12x^2 + 20x$ , which crosses the $x$ -axis at the origin $O$ and the points $A$ …
35The diagram shows part of the curve $y = \frac{4x + 1}{x - 2}$ . A line intersects the curve at points $P(11, 5)$ and $Q(3, 13)$ …
36The diagram shows part of the curve $y = \frac{8}{2x+3}$ . The tangent to the curve at the point $S\left(\frac{3}{2}, \frac{4}{3}\right)$ …
37曲线 $y=\cos x(0\leqslant x\leqslant \dfrac{3\pi }{2})$ 与坐标轴围成的面积是。