# Area Between Two Curves

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A common question is to find the area between two curves.

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## Both curves above x-axis

To find the area between the curves $y = f(x)$ and $y = g(x)$ and the lines $x = a$ and $x = b$

Use the rule: **Area** $= \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx}$

- where $y = f(x)$ is the
**upper**curve

- and $y = g(x)$ is the
**lower**curve.

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#### Why is it so??

Subtracting the shaded area in the second graph from the shaded area in the first graph will leave the area between the curves

Hence

… … Area $= \big( \text{Area below upper curve} \big) - \big( \text{Area below lower curve} \big)$

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… … … … $= \displaystyle{ \int\limits_{x=a}^{x=b} f(x) \, dx - \int\limits_{x=a}^{x=b} g(x) \; dx}$

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… … … … $= \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx}$

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## Both curves below x-axis

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Find the area bounded by the curves $y = f(x)$ and $y = g(x)$

Use the **same** rule (negatives will cancel)

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… … Area $= \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx}$

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**Note** When the question asks you to find the Area bounded by two curves and there are **no terminals** specified:

- you are expected to find the points of intersection between the two curves (simultaneous equations)
- then use the intersection points as the terminals.

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## Curve above and below x-axis

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If one of the curves is above the x-axis and the other curve is below the x-axis:

Use the **same** rule (negatives will cancel)

… … Area $= \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx}$

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## Curves cross each other

If the two curves cross each other within the domain, then **DO NOT** use the same rule:

- The definite integral over the entire domain will
**not**be equal to the area. - The second part will give "negative area" and this will cancel with the "positive area"

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To find the area shaded:

- Find the point of intersection
- calculate the area of the two sections separately
- add the two areas together

**Note:** For each section, make sure you use **Upper Curve** minus **Lower Curve**

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… … Area $= \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx} + \displaystyle{ \int\limits_{x=b}^{x=c} g(x) - f(x) \; dx}$

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