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Integration

If one of the bounds of an integral is positive and the other is negative, do we need to separate the integral at zero, or can we just solve normally?

For example, if we need to calculate an integral from x= -3 to x= 3, should we calculate the integral from x= -3 to x=0 and then from x=0 to x= 3?...(I know that we can separate it and multiply by 2 if we know that it is symmetric...but what if we don't know or are not sure of it's symmetry)?

Thank you in advance

Reply 1

blue_shift86

Original post by Artus

if you want to work out area, then yes, if you just want to integrate between the limits, then no you don't. Good morning btw :smile:

Reply 2

Artus

Original post by blue_shift86

if you want to work out area, then yes, if you just want to integrate between the limits, then no you don't. Good morning btw :smile:

Thanks for answering...
What do you mean by "just want to integrate"...doesn't integrating mean finding the area? Also...when you said "yes", do you mean that "yes" I need to separate them or "yes" I don't need to separate?

Reply 3

blue_shift86

Original post by Artus

if you have something like sin x, and u want to integrate from say 0 to 360, you'll notice that the 180 to 360 bit goes under the x-axis. I can't think of a C1 example so using this one but it's the principle.

If you want to find the area between the curve and the x-axis, you have to separate it.

If you didn't separate the above example the integration would cancel itself out and would give you 0. Similarly if you want to find the area of a curve, if half of the curve goes under the x-axis, you will cancel out half of the area cos you'll have the minusing effect from the integration between the positive and negative limit.

Since it seems you are being asked just to integrate, i'd say stick to keeping the limits as they are. If it does ask to work out the area, u must split it into two where the graph goes under the x- axis. that step is VERY important.

Also, if the question doesn't say what to do, you do integration without separation, and your teacher says it's wrong, you can tell your teacher: "but the question doesn't explicitly say to find the area" :biggrin:

. Then u get brownie points. haha

Reply 4

marcusmerehay

Original post by blue_shift86

if you want to work out area, then yes, if you just want to integrate between the limits, then no you don't. Good morning btw :smile:

Not strictly true.

Consider the graph of

f(x)=sin(x-\pi)

and what happens if you separate the integration specifically at 0.

Artus, it is perfectly legitimate to integrate between -3 and 3, whatever the function, as long as the function is actually integrable.

What blue_shift was trying to explain was if you have to work out the area bounded between the x-axis and some function, (in which case you need to find the roots of the function and separate the integration at these points) though I don't think this is what you were getting at.

Integration

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