y =dy/dx 2x x2x2 3x 2 x3x3 x 2 + 2x 8x x 3 + 3x 2 + x 4 5x 4 y =dy/dx x x x x x 3 + x 2 + 2x 5x 4 + x x -2 + x -3 3x x x x

Integration AIMS: To recognise integration as the reverse of differentiation To be able to integrate various expressions. To recognise the integration notation.

Integration Integration is the reverse of differentiation. To differentiate you multiply by the power then lower the power by one. To integrate you raise the power by one then divide by the power. y = Multiply by the power Take 1 off the power dy/dx y = Divide by the power Add 1 to the power dy/dx

Integration

Why a constant of integration? When you differentiate the constant does not affect dy/dx. The information about the constant is lost. So, when you integrate we have to put a constant back on. The only problem is we don’t know what it is! Hence we end up with the constant on integration +c

Task Hexagon Puzzle. How to integrate poster –Make a poster describing how to integrate. –Make sure to include examples and any useful details.

Definite Integration AIMS: To be able to calculate a definite integral. To know how this links to the area under a curve. To be able to calculate the area under a curve.

Definite Integration Definite integration is where limits are given to be used. This allows you to actually calculate a value for the integral. You do not need to use +c Also related to the area under a curve.

Examples

Step by Step Step 1 – Integrate the function after the integral sign. Step 2 – Substitute the upper limit in to this. Step 3 – Subtract from this the lower limit substituted in. Step 4 – Carry out the calculations.

Definite Integration

Try some questions!

Correct Me Look carefully at the definite integration calculations. Each calculation contains some mistakes. Spot the mistakes and then correct the working to give the correct answer.

Corrections

Finding the Area Under a Curve Aims: To be able to use definite integration to find areas under curves. To be able to find the area between a curve and a line.

Area Under Curves Finding a definite integration is related to the area under a curve. If you take the two limits to be the x values then you get the area under the curve enclosed.

Example Find the area under the curve y = x between x = 1 and x = 3

Worksheet Questions Use integration to try to find the area for each graph. You may need to calculate the x-intercepts first to use as the limits.

Questions to try.