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The integral in question

The homework is already turned in so I won't be getting any credit changing it now but my instructor is adamant about the answer to part B. I just want some confirmation and justification as to what the correct answer is.

As the problem states, the area shaded in blue is represented by A. For part B of the problem, I answered it with A but I got a wrong answer. I asked my instructor about this and he said that the answer is '-A' because the area is under the x-axis, thus needing a '-' to make it a negative number. However, I think that it is wrong. I think the answer should be 'A' because the area, as a a whole which is a negative number, is already represent by 'A' so therefore, it shouldn't need a negative in front of it.

Is my thinking incorrect?

000
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  • There is a difference between the integral of a function and the area under the function. The area under the function equals the integral of the absolute value of the function. – nippon Apr 10 '16 at 19:10
  • There are two possible points of view. One can view area below the axis as negative. I think (and your instructor agrees) that this is not reasonable, that area is inherently non-negative. But the integral for sure is negative. So I would say that if the area is $A$, the integral is $-A$. – André Nicolas Apr 10 '16 at 19:12
  • @AndréNicolas Why is it correct to think of the area as inherently non-negative? – 000 Apr 10 '16 at 19:14
  • If you want to paint it red, you will need a positive amount of paint. More abstractly, the area of a region is independent of its location. If you push the region up by $50$, its area should not change. – André Nicolas Apr 10 '16 at 19:28

1 Answers1

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Sometimes, in elementary calculus courses, area below the $x$-axis is taken to be negative.

But very much more often, one takes the point of view that the area of a region ought to be independent of its location, that congruent regions have the same area. Thus the area $A$ of your region should be unchanged if we move the region up by $50$, putting it fully above the $x$-axis. To put it another way, the area of a region should remain unchanged if the $x$-axis is moved.

But the integral of the post is definitely negative, and its value is $-A$.

André Nicolas
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