What's the difference between signed area under a curve (when finding definite integral) and area between 2 curves (area is not negative here even when below x axis..why?) How do I know when to use signed area or not? Is it only for UNDER the curve, not BETWEEN?
I have my final in two days, so any help is appreciated! :]
this depends on how you calculate the difference!
If you calculate the integral of a function that lies under the x-axis,
the integration process sums an infinite series of negative f(x) values.
Hence if the function lies completely below the x-axis,
your integral will return a negative value.
If you want the area, you need to integrate the modulus of f(x)
so you just change the sign of your negative result.
You must however know that the graph does not cross the x-axis between
the endpoints you are integrating between, so that you know the function
is always negative between the bounds.
If however, you subtracted the function from the x-axis (f(x)=0) and integrated,
then you'd get the area as the x-axis is always greater than f(x) in this case.
Hence if a function always lies underneath another,
then if you subtract the lower function from the upper one,
and then integrate, you will calculate the area between the curves,
but if you subtract the upper from the lower, you get a negative answer.
That works fine if the graphs do not cross between the bounds of integration.