Simply Checking an Answer in Calculus

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The integral \int_{9}^{4} (x - \frac{1}{\sqrt{x}}) dx was incorrectly computed due to the limits being reversed. The correct limits should be \int_{4}^{9} (x - \frac{1}{\sqrt{x}}) dx, which leads to the answer of \frac{61}{2} or 30.5. Several participants confirmed that the initial calculations were correct, but the error was identified in the order of the limits. The discussion highlights the importance of correctly setting limits in calculus problems. Ultimately, the user was reassured that their work was valid once the limits were corrected.
The Bob
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Hello all, again.

Simple question.

Compute \int_{9}^{4} (x - \frac{1}{\sqrt{x}}) dx

I did this:

\int_{9}^{4} (x - \frac{1}{x^{0.5}}) dx = \int_{9}^{4} (x - x^{-0.5}) dx

\left[ \frac{1}{2} x^2 - 2 x^{0.5} \right]_{9}^{4} = \left[ \frac{x^2}{2} - 2 \sqrt{x} \right]_{9}^{4}

(\frac{81}{2} - 6) - (\frac{16}{2} - 4) = \frac{81 - 12}{2} - \frac{16 - 8}{2} = \frac{69}{2} - \frac{8}{2} = \frac{61}{2} = 30.5

2 months of calculus and and still I am not sure of what I have done wrong. The answers in my book gives 10 \frac{2}{3}

So if it is right then can someone enlighten me to what I have done wrong please.

Cheers

The Bob (2004 ©)
 
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The Bob said:
Hello all, again.

Simple question.

Compute \int_{9}^{4} (x - \frac{1}{\sqrt{x}}) dx

I did this:

\int_{9}^{4} (x - \frac{1}{x^{0.5}}) dx = \int_{9}^{4} (x - x^{-0.5}) dx

\left[ \frac{1}{2} x^2 - 2 x^{0.5} \right]_{9}^{4} = \left[ \frac{x^2}{2} - 2 \sqrt{x} \right]_{9}^{4}

(\frac{81}{2} - 6) - (\frac{16}{2} - 4) = \frac{81 - 12}{2} - \frac{16 - 8}{2} = \frac{69}{2} - \frac{8}{2} = \frac{61}{2} = 30.5

2 months of calculus and and still I am not sure of what I have done wrong. The answers in my book gives 10 \frac{2}{3}

So if it is right then can someone enlighten me to what I have done wrong please.

Cheers

The Bob (2004 ©)
Third step...
x - x^-0.5 gives 1/2x^2 - 2x^0.5... hmmm...
Your work looks right to me.. maybe a book error?
K.
 
Last edited:
Pseudo Statistic said:
Third step...
x - x^-0.5 gives 1/2x^2 - 2x^0.5... hmmm...
Your work looks right to me.. maybe a book error?
K.
That is what I was hoping for. I know I make simply mistakes but now 4 people reckon it is right.

Cheers. Still though, can anyone else see anything wrong at all?

The Bob (2004 ©)
 
Nope,both your and the books answers are incorrect.My answer is -\frac{61}{2} and I'm sure of it,because even my old rusty Maple says it is so...:wink:

Daniel.
 
That's just weird.. even my calculator gives 30.5. :\
And yeah, I get the negative answer too... only mistake in his work I see now.
 
Oh nuts. I see what I did wrong. I have written the original limtis of the question the wrong way around. My fault (obviously). I'm not used to the Tex of calculus problems.

It should have been \int_{4}^{9} (x - \frac{1}{\sqrt{x}}) dx to give \frac{61}{2}

Cheers. At least I can go to college on Wednesday and now I am right. :biggrin:

The Bob (2004 ©)
 
It seemed pretty weird to me too,with the larger value being the inferior limit,but,hey,anything is possible in mathemetics,right...?:wink:

Daiel.
 
dextercioby said:
It seemed pretty weird to me too,with the larger value being the inferior limit,but,hey,anything is possible in mathemetics,right...?:wink:

Daiel.
Most things are possible but I am yet to see a larger, inferior limit too, Daiel. :wink:

The Bob (2004 ©)
 
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