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PROGRAMME F12 INTEGRATION
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Programme F12: 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
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Programme F12: 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
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Programme F12: 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:
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Programme F12: 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
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Programme F12: Integration
Standard integrals Just as with derivatives we can construct a table of standard integrals:
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Programme F12: 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
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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:
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Programme F12: 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
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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:
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Programme F12: 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
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Programme F12: Integration
Integration by partial fractions To integrate we note that so that:
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Programme F12: 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
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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
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Programme F12: 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
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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.
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Programme F12: Integration
Integration as a summation If the area is beneath the x-axis then the integral is negative.
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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
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