Form 5 Chapter 3: Integration (Pengamiran)
Total Marks: 40 Marks • Time Allowed: 50 Minutes
A Bahagian A: SPM Paper 1 Format [16 Marks]
Answer all questionsGiven that $\int_1^5 f(x)\,dx = 8$ and $\int_1^5 [2f(x) - kx]\,dx = 4$. Find the value of the constant $k$.
A curve has a second derivative $\frac{d^2y}{dx^2} = 6x - 4$. The tangent to the curve at the point $(1, 3)$ has a gradient of $5$. Find the equation of the curve.
Diagram below shows part of the curve $y = 3\sqrt{x}$ and the line $x = 4$.
(a) Calculate the area of the region bounded by the curve, the $y$-axis, and the line $y = 6$. [4 marks]
(b) Calculate the volume of revolution generated, in terms of $\pi$, when this region is rotated through $360^\circ$ about the $y$-axis. [4 marks]
B Bahagian B: SPM Paper 2 Format & Real-World KBAT [24 Marks]
Answer all questionsThe curve $y = 6x - x^2$ intersects the straight line $y = 2x$ at the origin $(0,0)$ and point $P$.
(a) Find the coordinates of point $P$. [2 marks]
(b) Calculate the area of the region enclosed between the curve and the straight line. [4 marks]
(c) Calculate the volume of solid generated when the region bounded by the curve $y = 6x - x^2$ and the $x$-axis from $x = 0$ to $x = 6$ is rotated through $180^\circ$ about the $x$-axis. [4 marks]
(a) Calculate the maximum radius of the bowl at the top rim ($y = 4\text{ m}$). [1 mark]
(b) Calculate the total water capacity volume, in $\text{m}^3$, of the fountain bowl. [3 marks]
(c) When the bowl is completely filled, a maintenance drainage pump empties the water at a constant rate of $0.5\pi\text{ m}^3/\text{min}$. Calculate the time, in minutes, taken to completely drain the bowl. [2 marks]
(d) Determine the rate of decrease of the water depth $h$ when the water level is $1\text{ meter}$ from the bottom. [2 marks]
Given that $\int_2^6 g(x)\,dx = 10$.
(a) Find the value of $\int_6^2 [3g(x) - 4]\,dx$. [3 marks]
(b) Given that $\int_2^k g(x)\,dx + \int_k^6 g(x)\,dx = \int_2^6 g(x)\,dx$ for any real constant $k$. If $\int_2^4 g(x)\,dx = 3$, find the value of $\int_4^6 [g(x) + 2x]\,dx$. [3 marks]