Can an area have negative portions?
Once in my lecture said that the area of a sine wave is negative in the 3rd and 4th quadrants while talking about integration. Is that possible?
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$\begingroup$When you are integrating a function it is often said you are finding the area under the function. When the function dips below the x-axis the area bounded is above the curve, so it is considered a negative area.
Now bare in mind this is a mathematical concept; in the real world area is a magnitude and is never negative. If you are using the integration to find the real world area of something you'll want all portions of the area to be positive. To do this you'll have to integrate the function in pieces, with each piece being a section above or below the x-axis, then take the absolute value of each piece.
So for example for $sin(x)$ between $0$ and $2\pi$ you would want to split it into two parts, from $0$ to $\pi$ because it's all above the x-axis, and from $\pi$ to $2\pi$ because it's all below the x-axis. Then you integrate and take the absolute value of each part before adding them together.
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