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Integration. Introduction. Integration is the reverse process of Differentiation Differentiating gives us a formula for the gradient Integrating can get us the formula for the curve, if we know the gradient function It can also be used to calculate the Area under a curve. I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
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1. Integration

2. Introduction • Integration is the reverse process of Differentiation • Differentiating gives us a formula for the gradient • Integrating can get us the formula for the curve, if we know the gradient function • It can also be used to calculate the Area under a curve

3. Teachings for Exercise 8A

4. Integration You can integrate functions of the form f(x) = axn where ‘n’ is real and ‘a’ is a constant Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment. If: If: If: Differentiating Integrating Function Function Multiply by the power Divide by the power Reduce the power by 1 Increase the power by 1 So integrating 2x should give us x2, but we will be unsure as to whether a number has been added or taken away Gradient Function Gradient Function 8A

5. Integration You can integrate functions of the form f(x) = axn where ‘n’ is real and ‘a’ is a constant Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment. Mathematically speaking… If: Integrating Then: Function Divide by the power We increased the power by 1, then divided by the (new) power Increase the power by 1 Gradient Function 8A

6. Integration You can integrate functions of the form f(x) = axn where ‘n’ is real and ‘a’ is a constant Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment. Example Questions Integrate the following: a) Increase the power by one, and divide by the new power DO NOT FORGET TO ADD C! Integrating Function If: Divide by the power b) Increase the power by one, and divide by the new power DO NOT FORGET TO ADD C! Increase the power by 1 Then: Gradient Function 8A

7. Integration You can integrate functions of the form f(x) = axn where ‘n’ is real and ‘a’ is a constant Integrating is the reverse process of differentiation. Let us think about a differentiation for a moment. Example Questions Integrate the following: c) Increase the power by one, and divide by the new power DO NOT FORGET TO ADD C! Integrating Function If: Divide by the power d) Increase the power by one, and divide by the new power DO NOT FORGET TO ADD C! Increase the power by 1 Then: Gradient Function 8A

8. Teachings for Exercise 8B

9. Integration You can apply the idea of Integration separately to each term of dy/dx  In short, if you have multiple terms to integrate, do them all separately Example Question Integrate the following: Integrate each part separately ‘Tidy up’ terms if possible 8B

10. Integration You can apply the idea of Integration separately to each term of dy/dx  In short, if you have multiple terms to integrate, do them all separately Example Question Integrate the following: Integrate each part separately Deal with the fractions Rewrite if necessary 8B

11. Teachings for Exercise 8C

12. Integration You need to be able to use the correct notation for Integration Example Question Find: Integrate each part separately This the the integral sign, meaning integrate The dx is telling you to integrate ‘with respect to x’ Deal with the fractions This is the expression to be integrated (brackets are often used to separate it) 8C

13. Integration You need to be able to use the correct notation for Integration Example Question Find: Integrate each part separately This the the integral sign, meaning integrate The dx is telling you to integrate the ‘x’ parts Deal with the fractions This is the expression to be integrated (brackets are often used to separate it) 8C

14. Integration You need to be able to use the correct notation for Integration Example Question Find: Integrate each part separately This the the integral sign, meaning integrate The dx is telling you to integrate the ‘x’ parts Deal with the fractions This is the expression to be integrated (brackets are often used to separate it) p and q2 should be treated as if they were just numbers! 8C

15. Integration You need to be able to use the correct notation for Integration Example Question Find: Integrate each part separately This the the integral sign, meaning integrate The dx is telling you to integrate the ‘x’ parts This is the expression to be integrated (brackets are often used to separate it) 8C

16. Teachings for Exercise 8D

17. Teachings for Exercise 8E

18. Integration Example Question You can find the constant of integration, c, if you are given a point that the function passes through Up until now we have written ‘c’ when Integrating. The point of this was that if we differentiate a number on its own, it disappears. Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one… Step 1: Integrate as before, putting in ‘c’ Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. The curve X with equation y = f(x) passes through the point (2,15). Given that: Find the equation of X. Integrate Sub in (2,15) Work out each fraction Add the fractions together Work out c 8E

19. Integration You can find the constant of integration, c, if you are given a point that the function passes through Up until now we have written ‘c’ when Integrating. The point of this was that if we differentiate a number on its own, it disappears. Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one… Step 1: Integrate as before, putting in ‘c’ Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. Example Question The curve X with equation y = f(x) passes through the point (4,5). Given that: Find the equation of X. Split into 2 parts Write in the form axn Integrate 8E

20. Integration You can find the constant of integration, c, if you are given a point that the function passes through Up until now we have written ‘c’ when Integrating. The point of this was that if we differentiate a number on its own, it disappears. Consequently, when integrating, we cannot be sure whether a number was there originally, and what it was if there was one… Step 1: Integrate as before, putting in ‘c’ Step 2: Substitute the coordinate in, and work out what ‘c’ must be to make the equation balance.. Example Question The curve X with equation y = f(x) passes through the point (4,5). Given that: Find the equation of X. Rewrite for substitution y = 5, x = 4 Work out each part carefully 8E

21. Summary • We have learnt what Integration is • We have seen it combined with rewriting for substitution • We have learnt how to calculate the missing value ‘c’, and why it exists in the first place