Week 8 – SS2 Third Term Further Mathematics Notes

WEEK EIGHT
TOPIC : INTEGRATION [INDEFINITE INTEGRALDEFINITE INTEGRAL AND AREA UNDER CURVE]
The process of reversing differentiation is called Integration. If dy/dx = 3x², then y could be x³, as the derivative of x³is 3x².
We say that x³ is an integral of 3x² with respect to x. The symbol for integration sign is given by ∫ . The expression to be integrated is put between the ∫ sign and dx.
∫ 3×2 dx could be x³
Since differentiating any constant gives zero, the following also have derivation 3x².
X³ + 2, x³ + 4.5, x³ – 17etc
In general, any function of the form x3 + c, where c is the constant has derivative of 3x²
Hence, ∫ 3x² dx = x³ + C. C is called constant of integration. Because we do not know the actual or definite value of C, this is called INDEFINITE INTEGRAL.

 Let y = xⁿ⁺¹/ n+1 , Differentiating dy/dx = (n+1) xⁿ⁺¹/n+1 = xⁿ
Reversing this,
∫ xⁿ dx = xⁿ⁺¹/n+1 + C
∫ kxn dx = K ^xn+1/n+1 + C
To Integrate a sum, integrate each term as this is similar to differentiating a sum.
Examples:
Evaluate (i) ∫(2x³ + 3x² – 4) dx
    2/4x⁴ + 3/3x³ – 4x + C
    ½ x2 + x3 – 4x + C
Evaluate ∫(4t³ + 2t² + ½ t²) dt
4/4 t4 + 2/3 t3 + ½ × ½ t² + C
t⁴ + 2t3 + ¼ t2 + C.

 Integration of basic Trigonometric functions
Consider the table below for differentiating trigonometric functions.

 Consider a similar table as the one given above

 Example
∫(cos2x + sin3x) dx
∫cos2x dx + ∫ sin3x dx
½ sin2x + -1/3 cos3x
½ sin2x – 1/3 cos3x + K.
Sometimes,the value of the constant C can be found, if extra information is given.
If dy/dx = x2+2x-3, find y in terms of x given that x = 1, y =4
Y =∫(x2 +2x-3)dx
Y =x3/3 + 2x/2 – 3x + C
Y = x3/3 +x2 -3x + C
Putting x=1,and y = 4
4 = 1/3 + 1 – 3 + C. hence; C = 5 2/
;. 1/3×3 – x2 – 3x + 5 2/3.

 If dy/dx = 2Cos3x, find y given that y =2 and x = 1/6∏
∫2Cos3x = 2/3Sin3x + C
2 = 2/3Sin3(∏/6) + C = 2/3 Sin∏/2 + C
Multiplying by ½ ∏ (180/∏) = 90⁰
Sin ½ ∏ – Sin 90⁰ = 1
2 = 2/3 ×1 + C
C = 4/3
Y = 2/3Sin3x + 4/3 .

 Evaluation:
Evaluate these indefinite integrals

1. ∫(y2 – 7y)dy
2. ∫(3×2 – 2x – 1)dx
3. ∫(Cos4x)dx
4. ∫(3cos2x + 4sin3x)dx
5. If dy/dx = 4×2 + 1 and y = 2 when x =3, find y in terms of x
6. If dy/dx = 2Sin1/3x and y = 4 when x =∏m, find y

    DEFINITE INTEGRALS
   In this part, the constant is removed. If a definite integration is performed, the function is evaluated between the values called limits. Upper and lower ie
   Example: Evaluate
   =x³ + C , (3³ + c) – (2³ + c)
   (27 + C) – (8 + C)
   27 + C -8 – C
   19.
   = 19.
   = { ½ Sin2}
   (½ sin× 0)
   ½ sin
   = ½

    
    Area under curve using Definite Integral
   Given in the diagram, the area between the curve and the x axis from x = a and to x = b. The area is given by

    
    
    
    
    
    
    
     Area =
   
    The area can be explained as: Area = ∫ y × dx = Sum of height of rectangle × width of rectangle
   y
   
    
    
    
    
    
    
    
    
    a b
   
    Ex 1: Find the Area between the curve y = x3 – x and the x axis when x =2 and x = 4.
   y
   
    
    
    
    
    0 2 4 x

    
   { ¼ (4)⁴ – ½(4)2} – { ¼ X 2⁴ – ½ (2)²}

    64 – 8 -4 + 2 = 54 UNITS.

    Ex 2 : Find the area in the diagram shown below
   

    
    y = 4 – x²
   4

    
   -2 0 2
   ∫(4 – x2 )dx
   {4x – x3/3}₂².
   4(2) – 2³/3 – 4(-2) – (-2)³/3
   (8- 8/3) – (-8 + 8/3)
   8 – 8/3 + 8 – 8/3
   32/3.square unitrs

    Evaluation:
   Find the area between the values as shown below
   y
   
     y = 3x²
   
     0 2 4 x
   
    
    
    
    
    
    Find the area between the curve y = x² + 3 At the xs axis and when x = -1 and x = 3.

    Evaluation
   1. x² and the line y = 2.

    General Evaluation:
   

7. ∫(3x – 1)(x + 2) dx
8. ∫5cos4x (dx)
9.  Reading Assignment :Solve the evaluation questions given above

    Weekend Assignment:
   

10. Evaluate A. 2/3 B. -2/3 C. -6 2/3 D. 6 2/3
11. Evaluate A. 4 B. 2 C. 4/3 D. 1/3
12. Evaluate A. – ½ B. 1 C. -1 D. 0
13. Find the area enclosed by by the curve y = x² , X = 0 and X = 3 A. 9 B. 7 C. 5/2 D. 5
14. Given y = 3x -2, x=3, x=4. Find the area under the curve A. 4/3 B. 17/2 C. 6 D. 3

    Theory
    

15. Find the area enclosed between the curve y =x2 + x -2 and the x axis
16. Find the area enclosed by the curve y = x2 – 3x + 3 and the y = 1.