Integration

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

3.5 Integration as the Summation of Areas (A) Area of the region between a Curve and the x-axis. Area of the shaded region;  A = ∫ a b y d x (B) Area of the region between a curve and the y-axis. Area of the shaded region;  A = ∫ a b x d y … Read more

3.4c Integration as the Inverse of DifferentiationExample: Shows that  d dx [ 2x+5 x 2 −3 ]= −2( x 2 +5x+3 ) ( x 2 −3 ) 2 Hence, find the value of  ∫ 0 2 ( x 2 +5x+3 ) ( x 2 −3 ) 2  dx Solution: d dx [ 2x+5 x 2 −3 ]= ( x 2 … Read more

3.4b Laws of Definite Integrals Example: Given that ∫ 3 7 f ( x ) d x = 5 , find the values for each of the following: (a) ∫ 3 7 6 f ( x ) d x (b) ∫ 3 7 [ 3 − f ( x ) ] d x (c) ∫ … Read more

3.4a Definite Integral of f(x) from x=a to x=b Example: Evaluate each of the following.   (a) ∫ − 1 0 ( 3 x 2 − 2 x + 5 ) d x (b) ∫ 0 2 ( 2 x + 1 ) 3 d x Solution:   (a) ∫ − 1 0 ( 3 … Read more

3.3 Finding Equation of a Curve from its Gradient Function Example 1: Find the equation of the curve that has the gradient function  d y d x = 2 x + 8 and passes through the point (2, 3). Solution: y = ∫ ( 2 x + 8 ) y = 2 x 2 2 … Read more

3.2 Integration by Substitution It is given that ∫ ( a x + b ) n d x , n ≠ − 1.   (A) Using the Substitution method, Let u=ax+b Thus,  du dx =a    ∴dx= du a Example 1: ∫ ( 3x+5 ) 3 dx. Let u=3x+5     du dx =3 dx= du 3 ∫ ( … Read more

3.1 Integration as the Inverse of Differentiation, Integration of axn and integration of the Functions of the Sum/Difference of Algebraic Terms Type 1: ∫ a d x = a x + C Example ∫ 2 d x = 2 x + C Type 2: ∫ a x n d x = a x n + … Read more