Question

Need help understanding integrals. I'm not sure if I'm typing it correctly, but here's the question:
Find the definite integral 4 to 1 (x^2-4sqrtx)/x
Wolfram alpha keeps saying the answer is 1/2, but I keep getting -7.5. I'm pretty sure my answer is correct, but I'm never confident with my answers... is there some kind of rule when doing subtraction?

Answer 1

When you say integral 4 to 1 do you mean \[\int\limits_{4}^{1} or \int\limits_{1}^{4}\] ?

Answer 2

the second one

Answer 3

squiggly s with 4 on top :p

Answer 4

Oh ok then for future reference you should make sure you say "1 to 4," always say the bottom number to the top number.

Answer 5

Bad Religion, regurgitate. Indecision, it's not too late.

Answer 6

I'm sorry if a lot of my questions are a bit remedial, but the only way I can learn this stuff for sure is to have someone validate my thinking :p. haha. but I mean, I understand the concept that the integral is basically just the area under the curve.

Answer 7

All questions are good questions because it means you're learning something from them so never worry about sounding 'remedial.' I just wanted to be sure your question was interpreted correctly. So you can split up your integral into two integrals:\[\int\limits_{1}^{4} x ^{2}/x - \int\limits_{1}^{4} 4\sqrt{x}/x\] The first integral reduces down to just x, and the integral of x is 1/2x^2. Evaluated from 1 to 4,\[1/2(4)^{2} - 1/2(1)^{2} = 1/2(16)-1/2(1)=8-1/2=7.5\]
Then the second integral reduces down to just 4/sqrt(x) (do you know how?) or written as \\[\int\limits\limits_{1}^{4} 4x ^{-1/2} = 4*(2x ^{1/2})=8x ^{1/2}\] Evaluated from 1 to 4, that is: \[8(4)^{1/2} - 8(1)^{1/2} = 8(2)-8(1)=16-8=8\]
So the result from your first integral was 7.5 and the result from your second integral was 8. Your first integral minus your second integral = 7.5-8 = -1/2

Answer 8

ahh ok i see. So If I have substraction or addition, I split the equation into parts and take the definite integral of each part separately, right?

Answer 9

Correct :)

Answer 10

sweet :) Thank you so much for the help :D. I think math is one of the few subjects where you can actually get a definite answer to almost anything ;p

Answer 11

and it's not ridiculously subjective and biased...xD

Answer 12

No problem. Haha yes it can be satisfying to get a straight answer from math when so much of the rest of academia and the world itself is questionable :) Let me know if you need any more help with integrals!

Answer 13

thank you :). I will probably have a few more questions as soon as this stuff gets harder haha.