Why don't I get the correct answer when I set these two equations =

  • Thread starter rxh140630
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In summary, the author Apostol's formula for the integral of [t]^2 is true only when n is a natural number. The problem at hand cannot be solved by simply setting the two equations equal to each other. However, breaking up the integral and using Apostol's formula on the first part and directly evaluating the second can lead to a solution without blind plug and chugging.
  • #1
rxh140630
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Homework Statement
Find all x>0 for which $$\int_0^x [t]^{2} dt = 2(x-1) $$
Relevant Equations
The notation [x] denotes the greatest integer less than or equal to x
In the question above it, the author (Apostol) states: $$\int_0^n [t]^{2} dt = \frac{n(n-1)(2n-1)}{6}$$

Why can't I set the two equations = and get the result?

2(x-1) = x(x-1)(2x-1)/6 => 12 = 2x^2 - x => 0 = x^2-(x/2) -6

using quadratic equation I get the wrong answer
 
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  • #2
Apostol's formula is true when ##n## is a natural number. You aren't assuming that ##x## is a natural number, though.
 
  • #3
Infrared said:
Apostol's formula is true when ##n## is a natural number. You aren't assuming that ##x## is a natural number, though.
Ahh I see I see. Well how would you go about solving this problem without plug and chugging? Seems really hard
 
  • #4
Let ##n## be the floor of ##x##. I would try breaking up the integral over the intervals ##[0,n]## and ##[n,x]## (for example, ##\int_0^{7/2}=\int_0^3+\int_3^{7/2}##). You can use apostol's formula on the first piece, and directly evaluate the second.
 
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  • #5
Infrared said:
Let ##n## be the floor of ##x##. I would try breaking up the integral over the intervals ##[0,n]## and ##[n,x]## (for example, ##\int_0^{7/2}=\int_0^3+\int_3^{7/2}##). You can use apostol's formula on the first piece, and directly evaluate the second.

Still seems like it would involve plug and chugging, I think I'm just going to have to skip this problem sadly.
 
  • #6
There's a (very) small amount of case work, but no blind 'plug and chug' needed.
 
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  • #7
Infrared said:
There's a (very) small amount of case work, but no blind 'plug and chug' needed.

Guess I'll go back to it, lol.
 

Related to Why don't I get the correct answer when I set these two equations =

1. Why is the answer to my equation not what I expected?

There could be several reasons for this. It could be due to a mistake in the equation itself, incorrect input values, or using the wrong mathematical operation. It is important to double-check your equation and input values to ensure accuracy.

2. Why do I get an error message when I input my equations?

This could be due to a syntax error in your equation, such as using incorrect mathematical symbols or missing parentheses. Make sure to carefully check your equation for any mistakes and correct them before inputting it.

3. Why does my calculator give a different answer than my friend's calculator?

This could be due to differences in the calculator's programming or rounding errors. It is also possible that one of the calculators is using a different order of operations. To ensure accuracy, it is best to use a trusted and reliable calculator or double-check your calculations by hand.

4. Why is my answer different when I use a different method to solve the equation?

Different methods of solving equations can yield different results, especially when dealing with complex equations. It is important to use the correct method for the type of equation you are solving and to double-check your work for accuracy.

5. Why is my answer not a whole number?

Some equations may result in decimal or fraction answers, especially when dealing with division or square roots. This is a normal occurrence and does not necessarily mean the answer is incorrect. However, if a whole number answer is expected, it is important to check your equation and input values for accuracy.

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