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STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 INTEGRATION APPLICATIONS 1 PROGRAMME 19.

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Presentation on theme: "STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 INTEGRATION APPLICATIONS 1 PROGRAMME 19."— Presentation transcript:

1 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 INTEGRATION APPLICATIONS 1 PROGRAMME 19

2 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

3 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

4 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Areas under curves Definite integrals

5 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Areas under curves The area above the x-axis between the values x = a and x = b and beneath the curve in the diagram is given as the value of the integral evaluated between the limits x = a and x = b: where

6 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 If the integral is negative then the area lies below the x-axis. For example: Basic applications Areas under curves

7 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Definite integrals The integral with limits is called a definite integral:

8 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Definite integrals To evaluate a definite integral: (a)Integrate the function (omitting the constant of integration) and enclose within square brackets with the limits at the right-hand end. (b)Substitute the upper limit. (c)Substitute the lower limit (d)Subtract the second result from the first result.

9 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

10 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

11 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Parametric equations If a curve has parametric equations then: (a)Express x and y in terms of the parameter (b)Change the variable (c)Insert the limits of the parameter

12 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Parametric equations For example, if and then the area under the curve y Between t = 1 and t = 2 is:

13 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

14 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

15 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Mean values The mean value M of a variable y = f (x) between the values x = a and x = b is the height of the rectangle with base b – a and which has the same area as the area under the curve:

16 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

17 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values

18 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Root mean square (rms) values The root mean square value of y is the square root of the mean value of the squares of y between some stated limits:

19 STROUD Worked examples and exercises are in the text Programme 19: Integration applications 1 Learning outcomes Evaluate the area beneath a curve Evaluate the area beneath a curve given in parametric form Determine the mean value of a function between two points Evaluate the root mean square (rms) value of a function

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