Download Presentation
## PROGRAMME F12

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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 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 Constant of integration Integration is the reverse process of differentiation. For example: The integral of 4x3 is then written as: Its value is, however:**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**Standard integrals Just as with derivatives we can construct a table of standard integrals:**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 of polynomial expressions Just as polynomials are differentiated term by term so they are integrated, also term by term. For example:**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**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:**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 by partial fractions To integrate we note that so that:**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**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**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 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.**Programme F12: Integration**Integration as a summation If the area is beneath the x-axis then the integral is negative.**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:**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