1. STROUD Worked examples and exercises are in the text PROGRAMME F12 INTEGRATION

2. STROUD Worked examples and exercises are in the text 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

3. STROUD Worked examples and exercises are in the text 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

4. STROUD Worked examples and exercises are in the text Integration Constant of integration Programme F12: Integration Integration is the reverse process of differentiation. For example: The integral of 4x 3 is then written as: Its value is, however:

5. STROUD Worked examples and exercises are in the text 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

6. STROUD Worked examples and exercises are in the text Standard integrals Programme F12: Integration Just as with derivatives we can construct a table of standard integrals:

7. STROUD Worked examples and exercises are in the text 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

8. STROUD Worked examples and exercises are in the text Integration of polynomial expressions Programme F12: Integration Just as polynomials are differentiated term by term so they are integrated, also term by term. For example:

9. STROUD Worked examples and exercises are in the text 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

10. STROUD Worked examples and exercises are in the text Functions of a linear function of x Programme F12: Integration To integrate we change the variable by letting u = ax + b so that du = a.dx. Substituting into the integral yields:

11. STROUD Worked examples and exercises are in the text 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

12. STROUD Worked examples and exercises are in the text Integration by partial fractions Programme F12: Integration To integrate we note that so that:

13. STROUD Worked examples and exercises are in the text 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

14. STROUD Worked examples and exercises are in the text Areas under curves Programme F12: Integration Area A, bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b, is given by: where

15. STROUD Worked examples and exercises are in the text 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

16. STROUD Worked examples and exercises are in the text Integration as a summation Programme F12: Integration 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.

17. STROUD Worked examples and exercises are in the text Programme F12: Integration Integration as a summation If the area is beneath the x-axis then the integral is negative.

18. STROUD Worked examples and exercises are in the text Integration as a summation The area between a curve an intersecting line The area enclosed between y 1 = 25 – x 2 and y 2 = x + 13 is given as: Programme F12: Integration

19. STROUD Worked examples and exercises are in the text 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