1. Challenging problems
Area between curves

2. 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

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

4. 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
𝑥−𝑥 2 𝑑𝑥 − 𝑥 2 −10𝑥+24 𝑑𝑥 = 𝑥 2 −10𝑥+24 𝑑𝑥

5. = − 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)
= − −96 − −18+63−72
= 1 3

6. Practice 1
4) Find the area enclosed between 𝑦= 𝑥 2 −2𝑥 and 𝑦=3−4𝑥− 𝑥 2
5) Find the area enclosed between 𝑦= 𝑥 2 +2𝑥 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

7. = 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 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 −1
= 0 − − 2 3 −1
= 5 3

8. Practice 5
5) Find the area enclosed between 𝑦= 𝑥 2 +2𝑥 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 −2
= 16− − −16
= = 64 3

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

10. 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 –

11. END