Math topic中文

Definite integral area

definite integral as area under a curve

Practice questions

  1. 1The figure shows part of the curve y=g(x)y = g(x). (2,8)(2, 8) and (6,1)(6, 1) are two points on the curve. Given that ∫26y dx=32\int_{2}^{6} y \text{ } dx = 32…
  2. 2定积分∫01(2x−x2)dx\int_{0}^{1}(\sqrt{2x{-}x^{2}})dx等于()({ }) A. π4\dfrac{\pi}{4} B. π2\dfrac{\pi}{2} C. π−14\dfrac{\pi{-}1}{4}…
  3. 3The diagram shows part of the curve y=3x+4y = \sqrt{3x + 4}. The line ABAB is tangent to the curve at AA. The points BB an…
  4. 4The diagram shows part of the curve y=(1−x)(x−4)y = (1 - x)(x - 4) and two lines l1l_1 and l2l_2. The lines l1l_1 and l2l_2 are ta…
  5. 5∫−22(sin⁡x+4−x2)dx=()∫_{-2}^{2}(\sin x+\sqrt{4-x^{2}}){\rm d}x=(\quad) A. 44 B. 2π2π C. 4+2π4+2π D. 88
  6. 6The diagram shows part of the curve y=4−e−2xy = 4 - e^{-2x} which crosses the axes at AA and at BB. Find the coordinates of…
  7. 7The diagram shows part of the curve y=4x+5y = \sqrt{4x + 5}. The normal of the curve at PP meets the xx-axis at TT. The xx…
  8. 8The diagram shows part of the curve y=14x2−12ln⁡(x3)y = \frac{1}{4}x^2 - \frac{1}{2}\ln\left(\frac{x}{3}\right) and P(3, 2.25)P(3,\,2.25) is…
  9. 9The diagram shows the curve y=cos⁡x+sin⁡xy = \cos x + \sin x for 0≤x≤π0 \le x \le \pi. The curve meets the yy-axis at the point AA a…
  10. 10It is given that f(x)={4a2−x2for 0≤x≤2a,3x−6afor 2a<x<3a,f(x) = \begin{cases} \sqrt{4a^2 - x^2} & \text{for } 0 \le x \le 2a, \\ 3x - 6a & \text{for } 2a < x < 3a, \end{cases}…
  11. 11The diagram shows part of the curve y=76x+1y = \frac{7}{6x + 1} intersecting the yy-axis at A(0,7)A(0, 7). The tangent to the cu…
  12. 12已知曲线W1W_{1}：x2+y2=m2x^{2}+y^{2}=m^{2}，W2W_{2}：x4+y2=m2x^{4}+y^{2}=m^{2}，其中m>0.m>0. ①当m=1m=1时，曲线W1W_{1}与W2W_{2}有44个公共…
  13. 13The diagram shows part of the curve y=(9−x)(x−3)y=(9-x)(x-3) and the line y=k−3y=k-3, where k>3k>3. The line through the maximum point…
  14. 14The diagram shows part of the curve y=3x−2y = \sqrt{3x-2}. The normal to the curve at BB meets the xx-axis at CC. Given t…
  15. 15The diagram shows part of the curve y=39−2xy = \dfrac{3}{9-2x}. The point P (3, 1)P\ (3,\ 1) lies on the curve and the normal to th…
  16. 16The diagram shows part of the curve y=16+6x−x2y = 16 + 6x - x^2 intersecting the line y=26−xy = 26 - x at AA and BB. Find the coor…
  17. 17The diagram shows part of the curve y=1−sin⁡xy = 1 - \sin x passing through the point PP. The curve touches the xx-axis at th…
  18. 18The diagram shows part of the curve y=e(−12x+2)+x−1y = e^{\left(-\frac{1}{2}x + 2\right)} + x - 1 that intersects the yy-axis at RR…
  19. 19The curve y=ln⁡xy = \ln x cuts the xx-axis at PP. The area, AA units2^2, is enclosed by the curve, the xx-axis and the…
  20. 20The diagram below shows the curve y=x3−12x2+20xy = x^3 - 12x^2 + 20x, which crosses the xx-axis at the origin OO and the points AA…
  21. 21The diagram shows part of the curve y=1x2y = \frac{1}{x^2}. Find the area bounded by the curve y=1x2y = \frac{1}{x^2}, the lin…
  22. 22曲线y=cos⁡x(0⩽x⩽3π2)y=\cos x(0\leqslant x\leqslant \dfrac{3\pi }{2})与坐标轴围成的面积是 。