Question

Definite Integrals: This is stumping me!
Evaluate the definite integral (to three decimal places): (x-x^2)/(2(cuberoot(x)) dx on [-8,-1]

Answer 1

Ok, let me re-type this with math type:

Answer 2

\[\int\limits_{-8}^{-1} (x-x^2)/2\sqrt[3]{x} dx \]

Answer 3

I just always get nervous when there are factors in the denominator to integrate. I think I should try to bring them up with negative powers, but I'm not entirely sure how to go about that.

Answer 4

I think you can separate and integrate separately can't you?

Answer 5

Integrate separately? Do you mean that I should try to separate the terms?
something like this?
Int x/2(cuberoot x) - x^2/2(cuberoot x) ?
Int (x(2)^-1 (x)^-1/3) - (x^2 (2)^-1 (x)^-1/3) ?

Answer 6

@TuringTest May I have some help, please? >_< I'm stuck on this one and would love some guidance.

Answer 7

\[\int\limits_{-8}^{-1} (x-x^2)/2\sqrt[3]{x} dx=\int_{-8}^{-1}\frac12{x-x^2\over x^{1/3}}dx\]can you simplify that?

Answer 8

\[1/2 \int\limits_{-8}^{-1} x^{2/3} dx - 1/2 \int\limits_{-8}^{-1} x ^{5/3} dx \]
like this? o.o

Answer 9

I pulled the 1/2 out and separates x from x^2 and simplified the exponents, so I have two different terms to integrate. Is this correct?

Answer 10

you don't have to integrate the two separately, but that is correct

Answer 11

\[1/2 \int\limits_{-8}^{-1} x^{2/3} dx - 1/2 \int\limits_{-8}^{-1} x ^{5/3} dx=\frac12\int x^{2/3}-x^{5/3}dx\]

Answer 12

\[(3(8-5x)x^2)/(80\sqrt[3]{x}) \]
This is what I ended up getting. I did F(b)-F(a) and got 57.1125 as my answer. It was pretty complicated, but I can type my work if you'd like to look at it.

Answer 13

it shouldn't be that complicated... I don't know why you reverted back to having the cube root in the denominator

Answer 14

simply use\[\int x^ndx=\frac{x^{n+1}}{n+1}+C\]

Answer 15

Hm... let me check. I solved this problem two different ways, and that answer was from my first. I'll try what you just put.

Answer 16

so 1/2 [ (x^5/3)/5/3 - (x^8/3)/8/3]
let me simplify that
and then do F(b)-F(a)

Answer 17

yep :)

Answer 18

so far I have (-3/10 + 3/16) - (-9.6 + 48)

Answer 19

I've got -38.513 as my answer. O.o Very off from my first. I think I did this correctly, though. Let me check my work, but it sounds good to me.

Answer 20

evaluating this integral gives an imaginary answer, so I don't know exactly what they want you to do
there's not errors in the problem, right?\[\int_{-8}^{-1}{x-x^2\over 2\sqrt[3]x}dx\]

Answer 21

No errors that I know of. That is the problem exactly.

Answer 22

I didn't get an imaginary answer, so regardless, I must have done it incorrectly, huh? As to the imaginary answer, are you sure? I don't think it was a trick question.

Answer 23

I'm lost now... What should I do, just submit the work I have, try to prove it imaginary, or say that the definite integrable is undeterminable?

Answer 24

http://www.wolframalpha.com/input/?i=integral+from+-8+to+-1+of++%28x-x%5E2%29%2F%282x%5E%281%2F3%29%29+dx+

Answer 25

actually wolfram is probably just giving us the imaginary part and we don't need it, let me do it by hand....

Answer 26

1/2 [ (x^5/3)/5/3 - (x^8/3)/8/3] =\[\frac12[\frac35 x^{5/3}-\frac 38x^{8/3}]|_{-1}^{-8}\]

Answer 27

Ok thanks, and I'll check over my work. The problem most definitely said DEFINITE integral, but we usually don't work with imaginary numbers in our class, or with calculus in general, right?

Answer 28

no we shouldn't be dealing with imaginary numbers at this level of calc

Answer 29

they want the real roots, so in evaluation we want to take \[(-8)^{5/3}\] and the 1/3 in the exponent would cause an imaginary answer if you look at the big picture, but here we only take the real root, so \[(-8)^{5/3}=(-2)^5=-32\]

Answer 30

Ok I see, so 32 would be acceptable as an answer just dealing with real roots?

Answer 31

yes, but not as a final answer to the whole integral, I'm just showing you how to evaluate the fractional exponents without getting the imaginary part that wolfram gives

Answer 32

you still have to plug it all in

Answer 33

Right, let me try that.

Answer 34

I'll do the same, let me know what you get

Answer 35

But for F(b) it would still be the same right?
3(-1)/10 - 3(-1)/16
because it's just one to a power
or would the second term be (-1)^8/3 = (-1)^8 = 1 instead?
so it is 3(-1)/10 - 3(1)/16 ?

Answer 36

(-1)^8/3 = 1

Answer 37

So this is what I got:
(-3/10 - 3/16) - (3(-32)/10 - 3(-256)/16)
so far

Answer 38

I think that is right

Answer 39

wait, (-8)^(8/3) should be positive, since we raise to an even power at the end

Answer 40

oh so positive 256

Answer 41

right

Answer 42

so 57.1125?

Answer 43

I don't even own a calculator, so let me just write out the simplified form first

Answer 44

yeah seems right. I always stress about the idea we have made some small arithmetic error somewhere, but I got the same answer, so looks good

Answer 45

oh ok
well I have (-3/10 +3/16) - (-9.6 -48)
= -0.4875 + 57.6
= 57.1125
This is how I did it. I use a calculator periodically, kudos to you for not using one! You must be working very hard, so thank you.

Answer 46

I simplify it and then use google or wolfram, my TI-83 broke :P

Answer 47

... So 57.113 is what I got the first time I did this. -_- Looks like we came full circle, I guess. XD But I understand the problem much better now, so that's good.

Answer 48

I'm going to close this question and just look over my work to double-check. Thanks so much for investing so much time into helping me, I really appreciate it! You're getting a medal straightaway!

Answer 49

Thanks, but my reward is knowing I helped :)