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

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VCE Maths Methods - Approximating areas under curves

Area under curves

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• 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

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• 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

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• 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

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• 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

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• 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)

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Time (s)5 10 15 20

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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

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• 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

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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.