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
𝑦= 𝑥 2 +5𝑥+4
𝑦=𝑥+1
𝑥 2 + 𝑦 2 =13
𝑦=𝑥+1

3. 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!

4. 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

5. 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!

6. 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

7. 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 and 𝑦=𝑥+5
2) Find the area enclosed between 𝑦= 𝑥 2 −2𝑥 and 𝑦=5−𝑥
3) Find the area enclosed between 𝑦= 𝑥 2 −3𝑥 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

8. Practice 1
1) Find the area enclosed between 𝑦= 𝑥 2 +2𝑥 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 −2
= 2(1)− − 1 3 (1) 3 − 2(−2)− −2 2 − 1 3 (−2) 3
= 2− 1 2 − 1 3 − −4−2+ 8 3
= − − = 9 2

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

10. = 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 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
= 4− 8 3 − 0
= 4 3

11. 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 –

12. END