 ## 3.8.4 Integration, Long Questions (Question 7 & 8)

Question 7:Diagram below shows a curve y= 1 4 x 2 +3 which intersects the straight line y = x + 6 at point A.(a) Find the coordinates of A.(b) Calculate(i) the area of the shaded region M,(ii) the volume generated, in terms of π, when the shaded region N is revolved 360o about the … Read more

## 3.8.3 Integration, Long Questions (Question 5 & 6)

Question 5: In Diagram below, the straight line WY is normal to the curve   y = 1 2 x 2 + 1 at B (2, 4). The straight line BQ is parallel to the y–axis. Find (a) the value of t, (b) the area of the shaded region, (c) the volume generated, in terms … Read more

## 3.8.2 Integration, Long Questions (Question 3 & 4)

Question 3: The gradient function of a curve which passes through P(2, –14) is 6x² – 12x.  Find (a) the equation of the curve, (b) the coordinates of the turning points of the curve and determine whether each of the turning points is a maximum or a minimum. Solution: (a) Given gradient function of a curve  dy dx =6 x 2 −12x The equation of the curve, … Read more

## 3.8.1 Integration Long Questions (Question 1 & 2)

Question 1: A curve with gradient function 5 x − 5 x 2  has a turning point at (m, 9). (a) Find the value of m. (b) Determine whether the turning point is a maximum or a minimum point. (c) Find the equation of the curve. Solution: (a) d y d x = 5 x − … Read more

## 3.7.5 Integration, SPM Practice (Question 13)

Question 13: Given that y= x 2 2x−1 , show that dy dx = 2x( x−1 ) ( 2x−1 ) 2 . Hence, evaluate  ∫ −2 2 x( x−1 ) 4 ( 2x−1 ) 2  dx . Solution: y= x 2 2x−1 dy dx = ( 2x−1 )( 2x )−x( 2 ) ( 2x−1 ) 2     = 4 x 2 −2x−2 … Read more

## 3.7.4 Integration, SPM Practice (Question 9 – 12)

Question 9: Given y= 5x x 2 +1  and  dy dx =g( x ), find the value of  ∫ 0 3 2g( x )dx. Solution: Since dy dx =g( x ), thus y= ∫ g( x ) dx ∫ 0 3 2g( x )dx=2 ∫ 0 3 g( x )dx   =2 [ y ] 0 3   =2 [ 5x x 2 +1 … Read more

## 3.7.3 Integration, SPM Practice (Question 5 – 8)

Question 5: Given  ∫ ( 6 x 2 +1 )dx=m x 3 +x +c,  where m and c are constants, find (a) the value of m. (b) the value of c if  ∫ ( 6 x 2 +1 )dx=13 when x=1. Solution: (a) ∫ ( 6 x 2 +1 )dx=m x 3 +x +c 6 x 3 3 +x+c=m x 3 +x+c 2 x 3 +x+c=m x 3 +x+c Compare the both sides, ∴ m=2 (b) … Read more

## 3.7.2 Integration, Short Questions (Question 2 – 4)

Question 2: Given that  Given that ∫ 4 ( 1 + x ) 4 d x = m ( 1 + x ) n + c , find the values of m and n. Solution: ∫ 4 ( 1 + x ) 4 d x = m ( 1 + x ) n + c ∫ … Read more

## 3.7.1 Integration Short Questions (Question 1)

Question 1: Find the integral of each of the following . ( a ) ∫ ( 3 x 2 − 5 2 x 3 + 2 ) d x ( b ) ∫ x 2 ( x 5 + 2 x ) d x ( c ) ∫ 3 x 4 + 2 x x … Read more

## 3.6 Integration as the Summation of Volumes

3.6 Integration as the Summation of Volumes (1). The volume of the solid generated when the region enclosed by the curve y = f(x), the x-axis, the line x = a and the line x = b is revolved through 360° about the x-axis is given by V x = π ∫ a b y 2 … Read more