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8.4 Areas between curves and lines

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Syllabus:

I /We Do

Mini Plenary

You Do

Plenary

NNJR

LFQ

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Topic: Area between a curve and a straight line or another curve

LFQ: How do we find area between curve & straight line and/or between two curves?

WILFs:

1. Able to find the between a straight line and a curve. (B)

2. Able to find out the area after sketching the curve. (A/A*)

3. Able to determine the area between multiple curves.(A*)

Key Words:

Calculus,
differentiate,
integrate,
reverse,
indefinite,
definite,
constant,
evaluate,
intersection.

I /We Do

Mini Plenary

You Do

Plenary

NNJR

LFQ

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

You need to be able 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

y₁

y₂

Region R

a

b

x

y

🡪 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!

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

x

y

Example Question

Below is a diagram showing the equation y = x, as well as the curve y = x(4 – x). Find the Area bounded by the two lines.

y = x(4 – x)

y = x

R

1) Find where the lines cross (set the equations equal)

0

3

2) Integrate to find the Area

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

You need to be able 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

x

y

Example Question

Below is a diagram showing the equation y = x, as well as the curve y = x(4 – x). Find the Area bounded by the two lines.

y = x(4 – x)

y = x

R

1) Find where the lines cross (set the equations equal)

Expand the bracket

Subtract x

Factorise

0

3

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

You need to be able 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

x

y

Example Question

Below is a diagram showing the equation y = x, as well as the curve y = x(4 – x). Find the Area bounded by the two lines.

y = x(4 – x)

y = x

R

2) Integrate to find the Area

0

3

Expand and rearrange (higher equation – lower equation)

Integrate

Split and Substitute

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

x

y

Example Question

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.

y = x(x – 3)

y = 2x

R

0

A

1) Work out the coordinates of the major points..

B

2. Think about the Area you need to find

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

You need to be able 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

x

y

Example Question

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.

y = x(x – 3)

y = 2x

R

0

5

3

O

A

C

B

The Area we want will be The Area of Triangle OAB – The Area ACB, under the curve.

1) Work out the coordinates of the major points..

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

(5,10)

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

You need to be able 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

x

y

Example Question

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.

y = x(x – 3)

y = 2x

R

0

5

3

Area of Triangle OAB – The Area ACB

2) Area of the Triangle…

(5,10)

25

Substitute values in

Work it out!

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WILF: Able to find the between a straight line and a curve. (B)

Mini Plenary

You Do

Plenary

NNJR

I/We Do

LFQ

You need to be able 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

x

y

Example Question

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.

y = x(x – 3)

y = 2x

R

0

5

3

Area of Triangle OAB – The Area ACB

3) Area under the curve

(5,10)

25

Expand Bracket

Integrate

Split and Substitute

-

²⁶/₃

16 ¹/₃

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Areas between curves and lines

Find the Point of Intersection

Area under the curve
Area of the triangle
Shaded Area

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Areas between curves and lines

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A Harder One

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A Harder One

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Test Your Understanding

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Test Your Understanding

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Area Between A Curve & A Line

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Area Between A Curve & A Line

Determine points of intersection so that we have the bounds of the integral.

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Area Between A Curve & A Line

Determine points of intersection so that we have the bounds of the integral.

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Area Between A Curve & A Line

We could start with the area under the curve between the points of intersection…

…and then subtract the area of the trapezium/trapezoid under the straight line.

We could start with the area under the curve between the points of intersection…

✁

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Area Between A Curve & A Line

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Area Between A Curve & A Line

We could start with the area under the curve between the points of intersection…

…and then subtract the area of the trapezium/trapezoid under the straight line.

✁

Determine points of intersection so that we have the bounds of the integral.

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Area Between A Curve & A Line

We could start with the area under the curve between the points of intersection…

…and then subtract the area of the trapezium/trapezoid under the straight line.

✁

Determine points of intersection so that we have the bounds of the integral.

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Test Your Understanding

525k

drfrost.org/s/

Note that is an alternative method that involves first subtracting the two functions before integrating, which we will explore.

?

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Test Your Understanding

Note that is an alternative method that involves first subtracting the two functions before integrating, which we will explore.

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Further Harder Example

What areas should we subtract this time?

Start with the area of the triangle and subtract the area under the curve within it.

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