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PROGRAMME F12. INTEGRATION. Programme F12: Integration. Integration Standard integrals Integration of polynomial expressions Functions of a linear function of x Integration by partial fractions Areas under curves Integration as a summation. Programme F12: Integration. Integration

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slide1

PROGRAMME F12

INTEGRATION

slide2

Programme F12: Integration

Integration

Standard integrals

Integration of polynomial expressions

Functions of a linear function of x

Integration by partial fractions

Areas under curves

Integration as a summation

slide3

Programme F12: Integration

Integration

Standard integrals

Integration of polynomial expressions

Functions of a linear function of x

Integration by partial fractions

Areas under curves

Integration as a summation

slide4

Programme F12: Integration

Integration

Constant of integration

Integration is the reverse process of differentiation. For example:

The integral of 4x3 is then written as:

Its value is, however:

slide5

Programme F12: Integration

Integration

Standard integrals

Integration of polynomial expressions

Functions of a linear function of x

Integration by partial fractions

Areas under curves

Integration as a summation

slide6

Programme F12: Integration

Standard integrals

Just as with derivatives we can construct a table of standard integrals:

slide7

Programme F12: Integration

Integration

Standard integrals

Integration of polynomial expressions

Functions of a linear function of x

Integration by partial fractions

Areas under curves

Integration as a summation

slide8

Programme F12: Integration

Integration of polynomial expressions

Just as polynomials are differentiated term by term so they are integrated, also term by term. For example:

slide9

Programme F12: Integration

Integration

Standard integrals

Integration of polynomial expressions

Functions of a linear function of x

Integration by partial fractions

Areas under curves

Integration as a summation

slide10

Programme F12: Integration

Functions of a linear function of x

To integrate

we change the variable by letting u = ax + b so that du = a.dx. Substituting into the integral yields:

slide11

Programme F12: Integration

Integration

Standard integrals

Integration of polynomial expressions

Functions of a linear function of x

Integration by partial fractions

Areas under curves

Integration as a summation

slide12

Programme F12: Integration

Integration by partial fractions

To integrate we note that

so that:

slide13

Programme F12: Integration

Integration

Standard integrals

Integration of polynomial expressions

Functions of a linear function of x

Integration by partial fractions

Areas under curves

Integration as a summation

slide14

Programme F12: Integration

Areas under curves

Area A, bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b, is given by:

where

slide15

Programme F12: Integration

Integration

Standard integrals

Integration of polynomial expressions

Functions of a linear function of x

Integration by partial fractions

Areas under curves

Integration as a summation

slide16

Programme F12: Integration

Integration as a summation

Dividing the area beneath a curve into rectangular strips of width x gives an approximation to the area beneath the curve which coincides with the area beneath the curve in the limit as the width of the strips goes to zero.

slide17

Programme F12: Integration

Integration as a summation

If the area is beneath the x-axis then the integral is negative.

slide18

Programme F12: Integration

Integration as a summation

The area between a curve an intersecting line

The area enclosed between y1 = 25 – x2 and y2 = x + 13 is given as:

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Programme F12: Integration

Learning outcomes

  • Appreciate that integration is the reverse process of differentiation
  • Recognize the need for a constant of integration
  • Evaluate indefinite integrals of standard forms
  • Evaluate indefinite integrals of polynomials
  • Evaluate indefinite integrals of ‘functions of a linear function of x’
  • Integrate by partial fractions
  • Appreciate the definite integral is a measure of an area under a curve
  • Evaluate definite integrals of standard forms
  • Use the definite integral to find areas between a curve and the horizontal axis
  • Use the definite integral to find areas between a curve and a given straight line