1. INTEGRATION APPLICATIONS 1
PROGRAMME 18
INTEGRATION APPLICATIONS 1

2. Programme 18: Integration applications 1
Basic applications
Parametric equations
Mean values
Root mean square (rms) values

3. Programme 18: Integration applications 1
Basic applications
Parametric equations
Mean values
Root mean square (rms) values

4. Programme 18: Integration applications 1
Basic applications
Areas under curves
Definite integrals

5. Programme 18: 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. Programme 18: Integration applications 1
Basic applications
Areas under curves
If the integral is negative then the area lies below the x-axis. For example:

7. Programme 18: Integration applications 1
Basic applications
Definite integrals
The integral with limits
is called a definite integral:

8. Programme 18: Integration applications 1
Basic applications
Definite integrals
To evaluate a definite integral:
Integrate the function (omitting the constant of integration) and enclose within square brackets with the limits at the right-hand end.
Substitute the upper limit.
Substitute the lower limit
Subtract the second result from the first result.

9. Programme 18: Integration applications 1
Basic applications
Parametric equations
Mean values
Root mean square (rms) values

10. Programme 18: Integration applications 1
Parametric equations
If a curve has parametric equations then:
Express x and y in terms of the parameter
Change the variable
Insert the limits of the parameter

11. Programme 18: Integration applications 1
Parametric equations
For example, if x = a sin and y = b cos then the area under the curve y
between  = 0 and  =  is:

12. Programme 18: Integration applications 1
Basic applications
Parametric equations
Mean values
Root mean square (rms) values

13. Programme 18: 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:

14. Programme 18: Integration applications 1
Basic applications
Parametric equations
Mean values
Root mean square (rms) values

15. Programme 18: 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:

16. Programme 18: 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