The net area and total area involving definite integrals

[chwala]

TL;DR Summary

I was looking at the problem below;

Calculate $$\int_{-\frac{π}{4}}^{\frac{π}{4}} 4 \sin (2x) dx =0$$

Now this is pretty clear to me.

My confusion stems from literature, which i have been interpreting any limits given as to finding area under the curve, in this context $4 \sin (2x)$being our function and the limits being $x_1={-\dfrac{π}{4}}$ to $x_2={\dfrac{π}{4}}$.

on checking, i am informed that there is net area and total area. I would like more clarity on this. In other words is calculating the definite integral equivalent to calculating net area? as opposed to calculating the area under the curve (total area)? how do we distinguish the two?

Thanks

 

[anuttarasammyak]

I suppose :
One of them counts in a way that area above x axis has plus signature and area below x axis has minus signature. The other counts all of them plus. The former applies for integration. Due to my poor English I am not sure which is net and which is total.

 

[fresh_42]

Integration measures volume, yes. However, integrals are an oriented quantity. The coordinate axes have a direction, making a difference whether we go from left to right or from right to left. This difference is measured by the sign, such that we get positive and negative volumes. Hence, if we want to calculate what you call net volume, we must respect the mathematically given orientation and correct it before or after the integration.

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The hidden insights about it involve the Graßmann algebra of differential forms, determinants, the cross product, or the simple fact that minus times plus is minus as indicated in the picture.