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Can anyone explain this to me? What does if mean that the area may sometimes be negative but that the area must be positive??

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- #1

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Can anyone explain this to me? What does if mean that the area may sometimes be negative but that the area must be positive??

- #2

Simon Bridge

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This means you have to be careful about when the areas under the x-axis are subtracted.

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$$\int_a^b f(x) dx = A_1 - A_2$$

where ##A_1## is the area between the ##x## axis and the positive part of the function, and ##A_2## is the area between the ##x## axis and the negative part of the function.

In the picture in your attachment, area ##A_1## is approximated by the first four and last two rectangles, and ##A_2## is approximated by the middle five rectangles.

[edit]: By pitiful "explanation" I mean of course the one in your image attachment, not in the response given by Simon Bridge.:tongue:

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HallsofIvy

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A =\int_a^b y_{upper}-y_{lower}~dx$$Students get used to working problems where ##f(x)\ge 0## on ##[a,b]## with ##\int_a^bf(x)~dx## which works since ##y_{upper}=f(x)## and ##y_{lower}=0##, the ##x## axis. But when you are calculating the area between a curve and the ##x## axis, and the curve crosses the ##x## axis, ##x## axis is now the upper curve, and on that portion ##y_{upper}-y_{lower}## becomes ##0-f(x)##, which changes the sign of the integrand on that portion. You normally break such integrals into two or more parts to work them.

- #6

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For example:

$$ \int^{1}_{0}-x^{2}dx=\left[-\dfrac{1}{3}x^{3}\right]^{1}_{0} $$

##= -\dfrac{1}{3} ##

This indicates, that between x=0 and x=1 for x

For finding the area, for example, as mentioned above, floor-space, you take the absolute value of the integral. That is:

$$ \left|\int^{1}_{0}-x^{2}dx\right|=\left|\left[-\dfrac{1}{3}x^{3}\right]^{1}_{0}\right| $$

##=\left|-\dfrac{1}{3}\right| ##

##=\dfrac{1}{3}## units

- #7

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Remember to split your integral from [a, b] into smaller" pieces if there are sharp(cusps) or the curves where the area is bounded by intersect. Your book should explain. I believe there are 4 scenarios from the top of my head.

Would you like an example?

- #8

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Instead of the taking the absolute value. I believe the first option is best because it allows you to get an intuitive feel for the application aspects. Such as finance, bio, Chem problems.

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