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

WB15a Find the area enclosed by these two curves 𝑦= 𝑥 2 −10𝑥+24 and 𝑦= 4𝑥−𝑥 2 Sketch the graphs to start!

WB15b Find the area enclosed by these two curves 𝑦= 𝑥 2 −10𝑥+24 and 𝑦= 4𝑥−𝑥 2 Sketch the graphs to start! First find the intersection points A and B 𝑥 2 −10𝑥+24= 4𝑥−𝑥 2 The intersections points on the graphs are where 𝑥=3 and 𝑥=4 2 𝑥 2 −14𝑥+24=0 𝑥 2 −7𝑥+12=0 𝑥−4 𝑥−3 =0 Points (3, 3) and (4, 0) 𝑥=3 𝑜𝑟 𝑥=4 Second find the difference between the curves – make sure this is done in the right order 4𝑥−𝑥 2 − 𝑥 2 −10𝑥+24 =−2 𝑥 2 +14𝑥−24 3 4 4𝑥−𝑥 2 𝑑𝑥 − 3 4 𝑥 2 −10𝑥+24 𝑑𝑥 = 3 4 𝑥 2 −10𝑥+24 𝑑𝑥

= − 2 3 (4) 3 +7 (4) 2 −24(4) − − 2 3 (3) 3 +7 (3) 2 −24(3) WB15c Find the area enclosed by these two curves 𝑦= 𝑥 2 −10𝑥+24 and 𝑦= 4𝑥−𝑥 2 Sketch the graphs to start! Third – Integrate ! = − 2 3 𝑥 3 +7 𝑥 2 −24𝑥 4 3 3 4 𝑥 2 −10𝑥+24 𝑑𝑥 = − 2 3 (4) 3 +7 (4) 2 −24(4) − − 2 3 (3) 3 +7 (3) 2 −24(3) = − 128 3 +112−96 − −18+63−72 = 1 3

Practice 1 4) Find the area enclosed between 𝑦= 𝑥 2 −2𝑥+3 and 𝑦=3−4𝑥− 𝑥 2 5) Find the area enclosed between 𝑦= 𝑥 2 +2𝑥+2 and 𝑦=10+2𝑥− 𝑥 2 Solutions 4) Intersections (-1, 6) and (0, 3) Area = 5 3 5) Intersections (-2, 2) and (2, 10) Area = 64 3

= 5 3 4) Find the area enclosed between 𝑦= 𝑥 2 −2𝑥+3 and 𝑦=3−4𝑥− 𝑥 2 Practice 4 4) Find the area enclosed between 𝑦= 𝑥 2 −2𝑥+3 and 𝑦=3−4𝑥− 𝑥 2 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 −1, 6 𝑎𝑛𝑑 (0, 3) 3−4𝑥− 𝑥 2 − 𝑥 2 −2𝑥+3 =−2 𝑥 2 −2𝑥 −1 0 −2 𝑥 2 −2𝑥 = − 2 3 𝑥 3 −𝑥 2 0 −1 = 0 − − 2 3 −1 = 5 3

Practice 5 5) Find the area enclosed between 𝑦= 𝑥 2 +2𝑥+2 and 𝑦=10+2𝑥− 𝑥 2 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 −2, 2 𝑎𝑛𝑑 (2, 10) 10+2𝑥− 𝑥 2 − 𝑥 2 +2𝑥+2 =8− 𝑥 2 −2 2 8− 𝑥 2 = 8𝑥− 2 3 𝑥 3 2 −2 = 16− 16 3 − 16 3 −16 =21 1 3 = 64 3

https://undergroundmathematics.org/calculus-of-powers/meaningful-areas WB16

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 –

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