Why do you Tack on The Negative For this Integral

Thread starter bmed90
Start date Feb 7, 2012
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Integral Negative

In summary, the negative sign in front of an integral indicates that the area under the curve is being subtracted instead of added. It is not always necessary to include the negative sign, but it must be included for functions with both positive and negative values. The negative sign changes the sign of the result and can also be placed inside the integral to simplify calculations. Additionally, the negative sign may be included for other reasons such as matching a specific form or making the integral easier to evaluate. Careful consideration of the context and purpose of the integral is important when determining whether or not to include the negative sign.

Feb 7, 2012

bmed90

If you integrate

1/(1-y)dy

why do you end up with a negative in front of your answer

-ln|1-y|+c

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Feb 7, 2012

Char. Limit

Gold Member

Because using u-substitution, we get this:

u = 1 - y, du = - dy

And so...

[tex]\int \frac{1}{1-y} dy = - \int \frac{1}{u} du[/tex]

Integrating the right-hand side, we get - ln|u| + C, or - ln|1-y| + C.

Feb 7, 2012

bmed90

gotcha

1. Why do you tack on the negative for this integral?

The negative sign in front of an integral indicates that the area under the curve is being subtracted instead of added. This is because the integral is finding the signed area, which takes into account both positive and negative areas above and below the x-axis.

2. Is it always necessary to include the negative sign in an integral?

No, it is not always necessary. If the function being integrated is always positive, then the negative sign can be omitted. However, if the function has both positive and negative values, the negative sign must be included to accurately calculate the signed area.

3. How does the negative sign affect the value of the integral?

The negative sign changes the sign of the result, making it negative instead of positive. This is because the negative sign indicates a subtraction instead of addition. Without the negative sign, the integral would give the absolute value of the signed area instead of the signed value.

4. Can the negative sign be placed inside the integral?

Yes, the negative sign can be placed inside the integral as long as it is applied to the entire integrand. This is equivalent to placing the negative sign in front of the integral and can be used to simplify certain integrals.

5. Are there any other reasons for including the negative sign in an integral?

Yes, in some cases, the negative sign may be included to make the integral easier to evaluate or to match a specific form needed for a mathematical formula. It is always important to carefully consider the context and purpose of the integral when determining whether or not to include the negative sign.