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Integration

Integration. (Calculus Part 2). What is to be learned?. Relationship between integration and differentiation The Rule for Integration Some terminology and symbol stuff that we use. Remember Differentiation. y = 2x 3 dy / dx = 6x 2 Integration is Anti Differentiation

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Integration

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  1. Integration (Calculus Part 2)

  2. What is to be learned? • Relationship between integration and differentiation • The Rule for Integration • Some terminology and symbol stuff that we use

  3. Remember Differentiation y = 2x3 dy/dx = 6x2 Integration is Anti Differentiation dy/dx = 6x2 y = 2x3

  4. Integrating Find equation for y = 1. dy/dx = 5x4 2. dy/dx = 8x3 3. dy/dx = 6x 4. dy/dx = 17 y = x5 y = 2x4 y = 3x2 y = 17x

  5. Another Consideration y = x2 + 9 dy/dx = 2x y = x2 – 312 dy/dx = 2x so integrating dy/dx = 2x y = x2 - 312 + c + anything! +9 where c is a constant easy to forget!

  6. Establishing The Rule dy/dx = 8x3 ÷4 +1 y = 2x4

  7. Establishing The Rule dy/dx = 8x3 dy/dx = 20x4 ÷4 +1 y = 2x4 ÷5 +1 5 4x

  8. The Rule dy/dx = axn y = axn+1 +c n+1

  9. Examples dy/dx = 18x5 y = 18x6 y = 3x6 + c +c 6

  10. Examples dy/dx = 2x3 y = 2x4 y = ½ x4 + c or y = x 4 + c +c 4 2

  11. Examples dy/dx = 3x7 y = 3x8 +c 8

  12. Examples 1 dy/dx = 3x y = 3x2 +c 2

  13. Examples dy/dx = 7 y = 7x1 y = 7x + c Easiest just to remember this. x0 +c 1

  14. Integration The opposite of differentiation The Rule dy/dx = axn y = axn+1 +c n+1

  15. Ex 1 Differential Equation dy/dx = 24x3 y = 24x4 y = 6x4 + c +c 4

  16. Ex 2 dy/dx = 5x8 y = 5x9 +c 9

  17. Ex 3 dy/dx = 9 y = 9x + c This applies for any number

  18. Finding c If we are given extra information we can find c. Ex. dy/dx = 4x + 2 is the gradient of the tangent to a curve which passes through (1 , 8). Find the equation of the curve Integrating y = 2x2 + 2x + c 8 = 2(1)2 + 2(1) + c 8 = 4 + c c = 4 Equation is y = 2x2 + 2x + 4 x y

  19. Finding c If we are given extra information we can find c. Ex. dy/dx = 2x + 4 is the gradient of the tangent to a curve which passes through (2 , 10). Find the equation of the curve Integrating y = x2 + 4x + c 10 = (2)2 + 4(2) + c 10 = 12 + c c = -2 Equation is y = x2 + 4x – 2 x y

  20. Terminology (The Big S!) dx ∫ 4x + 2 = 2x2 + 2x ( ) + c

  21. New Rule – Same Difficulties ∫ √x dx = ∫ x dx = x = 2x ½ 3/2 ÷ 3/2  X2/3 3/2 3/2 + c 3

  22. New Rules – Same Difficulties 6 dx = ∫ 6x-3 dx = 6x = -3x ∫ x3 -2 -2 -2 + c

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