Areas between curves KUS objectives BAT use integration

Areas between curves • • • KUS objectives BAT use integration to find the area between a curve and the x -axis BAT use integration to find the area between a line and curve or between two curves Starter: solve the following simultaneous equations

Introduction To calculate the Area between a Curve and a Straight Line y Region R y 2 To work out the Region between 2 lines, you work out the region below the ‘higher’ line, and subtract the region below the ‘lower’ line y 1 a Sometimes you will need to work out the values of a and b Sometimes a and b will be different for each part MAKE SURE you put y 1 and y 2 the correct way around! b x

WB 13 Below is a diagram showing the equation y = x, as well as the curve y = x(4 – x). Find the Area bounded by the line and the curve. 1) Find where the lines cross (set the equations equal) y y=x Expand the bracket R Subtract x Factorise 0 3 x y = x(4 – x) 2) Integrate to find the Area Expand rearrange (higher equation – lower equation) Integrate Split and Substitute

WB 14 The diagram shows a sketch of the curve with equation y = x(x – 3), and the line with Equation 2 x. Calculate the Area of region R. 1) Work out the coordinates of the major points. . As the curve is y = x(x – 3), the x-coordinate at C = 3 y = x(x – 3) y y = 2 x A Set the equations equal to find the x-coordinates where they cross… Expand Bracket R Subtract 2 x Factorise O C B x 2) Area of the Triangle… Substitute values in Work it out! The Area we want will be The Area of Triangle OAB – The Area ACB, under the curve.

WB 14 continued The diagram shows a sketch of the curve with equation y = x(x – 3), and the line with Equation 2 x. Calculate the Area of region R. 3) Area under the curve Expand Bracket y = x(x – 3) y y = 2 x (5, 10) Integrate Split and Substitute 16 R 1/3 0 3 x 5 Area of Triangle OAB – The Area ACB 25 - 26/ 3

Practice 1

Practice 2

Practice 3

KUS objectives BAT use integration to find the area between a curve and the x-axis BAT use integration to find the area between a line and curve or between two curves self-assess One thing learned is – One thing to improve is –

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