Challenging problems Area between curves

Starter: solve the following simultaneous equations 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 𝑦= 𝑥 2 +5𝑥+4 𝑦=𝑥+1 𝑥 2 + 𝑦 2 =13 𝑦=𝑥+1

To calculate the Area between a Curve and a Straight Line Introduction To calculate the Area between a Curve and a Straight Line 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 Region R y2 y1 x a b  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 y1 and y2 the correct way around!

Expand and rearrange (higher equation – lower equation) WB13 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. y 1) Find where the lines cross (set the equations equal) y = x Expand the bracket R Subtract x Factorise x 3 2) Integrate to find the Area y = x(4 – x) Expand and rearrange (higher equation – lower equation) Integrate Split and Substitute

WB14 The diagram shows a sketch of the curve with equation y = x(x – 3), and the line with Equation 2x. Calculate the Area of region R. 1) Work out the coordinates of the major points.. x y y = x(x – 3) y = 2x R O A B C As the curve is y = x(x – 3), the x-coordinate at C = 3  Set the equations equal to find the x-coordinates where they cross… Expand Bracket Subtract 2x Factorise 2) Area of the Triangle… The Area we want will be The Area of Triangle OAB – The Area ACB, under the curve. Substitute values in Work it out!

Area of Triangle OAB – The Area ACB WB14 continued The diagram shows a sketch of the curve with equation y = x(x – 3), and the line with Equation 2x. Calculate the Area of region R. 3) Area under the curve Expand Bracket y = x(x – 3) y y = 2x (5,10) Integrate Split and Substitute 16 1/3 R x 3 5 Area of Triangle OAB – The Area ACB 25 - 26/3

1) Find the area enclosed between 𝑦= 𝑥 2 +2𝑥+3 and 𝑦=𝑥+5 Practice 1 1) Find the area enclosed between 𝑦= 𝑥 2 +2𝑥+3 and 𝑦=𝑥+5 2) Find the area enclosed between 𝑦= 𝑥 2 −2𝑥+3 and 𝑦=5−𝑥 3) Find the area enclosed between 𝑦= 𝑥 2 −3𝑥+5 and 𝑦=5−𝑥 Solutions Intersection points (-2, 3) (1, 6) Area = 4.5 Intersection points (-1, 6) (2, 3) Area = 9 2 Intersection points (0, 5) (2, 3) Area = 4 3

Practice 1 1) Find the area enclosed between 𝑦= 𝑥 2 +2𝑥+3 and 𝑦=𝑥+5 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 −2, 3 𝑎𝑛𝑑 (1, 6) 𝑥+5 − 𝑥 2 +2𝑥+3 =2−𝑥− 𝑥 2 −2 1 2−𝑥− 𝑥 2 = 2𝑥− 1 2 𝑥 2 − 1 3 𝑥 3 1 −2 = 2(1)− 1 2 1 2 − 1 3 (1) 3 − 2(−2)− 1 2 −2 2 − 1 3 (−2) 3 = 2− 1 2 − 1 3 − −4−2+ 8 3 = 7 6 − − 10 3 = 9 2

Practice 2 2) Find the area enclosed between 𝑦= 𝑥 2 −2𝑥+3 and 𝑦=5−𝑥 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 −1, 6 𝑎𝑛𝑑 (2, 3) 5−𝑥 − 𝑥 2 −2𝑥+3 =2+𝑥− 𝑥 2 −1 2 2+𝑥− 𝑥 2 = 2𝑥+ 1 2 𝑥 2 − 1 3 𝑥 3 2 −1 = 4+2− 8 3 − −2+ 1 2 + 1 3 = 10 3 − − 7 6 = 9 2

= 4 3 3) Find the area enclosed between 𝑦= 𝑥 2 −3𝑥+5 and 𝑦=5−𝑥 Practice 3 3) Find the area enclosed between 𝑦= 𝑥 2 −3𝑥+5 and 𝑦=5−𝑥 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 0, 5 𝑎𝑛𝑑 (2, 3) 5−𝑥 − 𝑥 2 −3𝑥+5 =2𝑥− 𝑥 2 0 2 2𝑥− 𝑥 2 = 𝑥 2 − 1 3 𝑥 3 2 0 = 4− 8 3 − 0 = 4 3

One thing to improve is – 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 –

END