PROGRAMME 18 INTEGRATION APPLICATIONS 1 STROUD Worked examples

PROGRAMME 18 INTEGRATION APPLICATIONS 1 STROUD Worked examples and exercises are in the text

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values STROUD Worked examples and exercises are in the text

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values STROUD Worked examples and exercises are in the text

Programme 18: Integration applications 1 Basic applications Areas under curves Definite integrals STROUD Worked examples and exercises are in the text

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 STROUD Worked examples and exercises are in the text

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: STROUD Worked examples and exercises are in the text

Programme 18: Integration applications 1 Basic applications Definite integrals The integral with limits is called a definite integral: STROUD Worked examples and exercises are in the text

Programme 18: 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. STROUD Worked examples and exercises are in the text

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values STROUD Worked examples and exercises are in the text

Programme 18: 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 STROUD Worked examples and exercises are in the text

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: STROUD Worked examples and exercises are in the text

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values STROUD Worked examples and exercises are in the text

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: STROUD Worked examples and exercises are in the text

Programme 18: Integration applications 1 Basic applications Parametric equations Mean values Root mean square (rms) values STROUD Worked examples and exercises are in the text

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: STROUD Worked examples and exercises are in the text

Programme 18: Integration applications 1 Learning outcomes üEvaluate the area beneath a curve given in parametric form üDetermine the mean value of a function between two points üEvaluate the root mean square (rms) value of a function STROUD Worked examples and exercises are in the text