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VCE Maths Methods - Approximating areas under curves
Approximating areas under curves
• Area under curves• Left end-point approximation• Right end-point approximation• Improvements to rectangle approximations• Find the total from a rate
1
VCE Maths Methods - Approximating areas under curves
Area under curves
2
• The area under a curve is useful on it’s own.
• The area under a curve can also be used to !nd the total amount of quantity from a known rate and time relationship.
• The area under the curve can be approximated using rectangles / triangles / trapeziums.
VCE Maths Methods - Approximating areas under curves
Left end-point approximation
3
• The heights of the rectangles are determined by the height at the left boundary of the rectangle.
• For curves with a positive gradient, this method will under-estimate the area.
A1 A2
A1=1×2=2
y =22
4+1=2
A2 =1×13
4=13
4 A≈2+13
4≈21
4≈5.25
y = 32
4+1=13
4
y = x 2
4+1
VCE Maths Methods - Approximating areas under curves
Right end-point approximation
4
• The heights of the rectangles are determined by the height at the right boundary of the rectangle.
• For curves with a positive gradient, this method will over-estimate the area.
A1 A2
y = 32
4+1=13
4
A2 =1×5=5 A≈13
4+5≈ 33
4≈8.25
A1=1×13
4=13
4
y = 42
4+1=5
VCE Maths Methods - Approximating areas under curves
Improvements to rectangle approximations
5
• Use narrower columns.
• Take an average of the left & right end point rectangles.
• Use the midpoint of the rectangle.
• Use a trapezium to calculate the area.
• These last two methods both produce an equivalent estimate of the area.
Trapezium area= 1
2(a+b )h
h
a
b
VCE Maths Methods - Approximating areas under curves
Find the total from a rate
6
• Often we need to !nd the total amount of a quantity from a time rate.
• For this we use a process, multiply the rate by the time gives the total.
• For example, a car travelling at 60 km/h.
• Travelling for two hours:
• This is a distance of 120 km (60 km/h x 2 h = 120 km).
VCE Maths Methods - Approximating areas under curves
Speed (m/s)
10
5
Time (s)5 10 15 20
15
Find the total from a rate
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• The area under a speed-time graph gives the distance covered.
• If the speed is changing, the area under the graph still gives the distance.
Red car: d = 9 m/s x 20 s = 180 m
Blue car: d = ½ x 12 m/s x 10s = 60 md = 12 m/s x 10s = 120 mTotal = 180 m
VCE Maths Methods - Approximating areas under curves
Find the total from a rate
8
• What if the rate was more erratic?
• We could approximate the area by breaking it up into smaller squares and !nd the total area.
Speed (m/s)
Time (s)
8.5m/s×1s= 8.5m
105 2015
10
15
5
Each square area on the graph represents a distance travelled:
1 m/s x 1 s = 1 m.
There are about 220 squares under the curve.
This represents a distance of 220 m.
The average speed was 220m/20s =11 m/s.
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