Integration Question: Confused on Limits of Integration

[nmsurobert]

im working on some homework that the instructor recently gave us the solutions to and I am confused on something that he's done.
intailly i have

∫N²(x+a/2)²(x-a/2)² the integral is from a/2 to -a/2

the next step is this

∫2N²(x²-a²/4)² integrated from a/2 to 0

i don't understand why the limits of integration are changing and where the new numbers are coming from. is this something that can be done because the initial limits are opposites of each other?

 

[andrewkirk]

$N^2(x^2-a^2/4)^2$ is an even function, so its integral from -a/2 to 0 is the same as its integral from 0 to a/2. So we just calculate one of them and double the result. That's where the coefficient of 2 in the second formula comes from.

 

[nmsurobert]

ahhh ok. i was thinking that but i couldn't find anything in the text about it so i wanted to make sure.
i suppose that's one of those things i should just know about by now haha

thank you mr andrewkirk!

 

[Mark44]

im working on some homework that the instructor recently gave us the solutions to and I am confused on something that he's done.
intailly i have

∫N²(x+a/2)²(x-a/2)² the integral is from a/2 to -a/2

the next step is this

∫2N²(x²-a²/4)² integrated from a/2 to 0

You really should get in the habit of including the differential with your integrals. When you're first learning about integrals, the 'dx' or whatever it happens to be seems superfluous (like a human appendix), but omitting it will come back around and bite you when you're working with more complicated substitutions such as trig substitution or integration by parts.

 

[nmsurobert]

You really should get in the habit of including the differential with your integrals. When you're first learning about integrals, the 'dx' or whatever it happens to be seems superfluous (like a human appendix), but omitting it will come back around and bite you when you're working with more complicated substitutions such as trig substitution or integration by parts.

totally agree with you. my paper is covered with them, i promise. i just forgot to put it in as i was typing it.
my calc2 instructor would bleed all over our assignments if we forgot to put those haha