08 4areabween

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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• 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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