6FAreaBetweenCurves


 * Area Between Two Curves**

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: math . \qquad \text{Area } \; = \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx} \qquad. math

where **//y = f(x)//** is the __upper__ curve

and **//y = g(x)//** is the __lower__ curve.

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 math . \qquad \text{ Area } = \big( \text{Area below upper curve} \big) - \big( \text{Area below lower curve} \big) \qquad. math . math . \qquad \qquad = \displaystyle{ \int\limits_{x=a}^{x=b} f(x) \, dx - \int\limits_{x=a}^{x=b} g(x) \; dx} \qquad. math . math . \qquad \qquad = \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx} \qquad. math

Both curves below x-axis

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

Use the __same__ rule (negatives will cancel) math . \qquad \text{Area } \; = \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx} \qquad. math

Note When there are no terminals specified you are expected to find and use the points of intersection between the two curves.

Curve above and below x-axis

Use the __same__ rule (negatives will cancel)

math . \qquad \text{Area } \; = \displaystyle{ \int\limits_{x=a}^{x=b} f(x) - g(x) \; dx} \qquad. math

Curves cross each other

Calculate the area of the two sections separately and add them together.

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

math . \qquad \text{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} \qquad. math

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