Integrating Definite Integral of (x - x^2)*(2x^(-1/3)) from -8 to -1

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The discussion focuses on the integration of the function (x - x^2)*(2x^(-1/3)) from -8 to -1. Participants identify errors in the original equation's formatting, particularly regarding the placement of the constant 2 in the denominator. The calculations reveal discrepancies in the final results, with one participant obtaining 57.112 while others point out mistakes in exponent handling and multiplication. The importance of careful notation and arithmetic in integration is emphasized, as participants express concern over making similar errors on tests. Overall, the thread highlights common pitfalls in calculus integration and the need for precision in mathematical expressions.
VikingStorm
INT[-8 to -1] x - x^2 / 2*x^(1/3) dx

(x - x^2)*(2x^(-1/3))

Distributed:
2x^(2/3) - 2x^(5/3)

[6x^(5/3) / 5] -[ 3x^(8/3) / 4]

Plug in -1, and -8

-1.2 - -.75 = -.45

-38.4 - - 192 = 153.6

-.45 - 153.6 = -154.05

When I put this into the calculator straight to check my work, I get 57.112 as the answer. What did I do wrong here?
 
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Originally posted by VikingStorm
x - x^2 / 2*x^(1/3) dx

(x - x^2)*(2x^(-1/3))

There is the error. In your original equation, the 2 is in the denominator. In your second equation, you brought the 2 to the top without putting a negative exponent on it; you only put a negative exponent on the x.
 
<br /> \frac{x-x^2}{2x^{1/3}}<br /> = (x-x^2) (\frac{1}{2} x^{-1/3})<br />

<br /> (-1)^{8/3} = 1<br />
 
Ah, I thought it was just a constant multiplier that stuck with the x.

Hmm...

So that would make it:
[ 2^-1 * x^(2/3)] - [2^-1 * x^(5/3)]

5x^(5/3)/6 - 3x^(8/3)/16

For -8, I get -26.7 - - 48 = 21.3

-1, -.833 - -.1875 = -.6455

Ay... I must have done something else wrong?
 
(-1)^{8/3} = 1

(and this time you multiplied by five-thirds instead of divided)
 
Urgh, hopefully I won't make these simple mistakes on the test.
 
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