My question for this week is about finding the area underneath a curve, which is

Question

My question for this week is about finding the area underneath a curve, which is supposed to be the "real world" application of the Fundamental Theorem of Calculus. So far, every "area" question or every definite integral that we have had to answer is ABOVE the x-axis. For example, section 2.4.4, #2 in the Thinkwell exercise book is to find the area between the curve y= x^3+3 and the x-axis between the interval [0,1]. If you graph this, you will see that the entire area between that interval lies above the x-axis, so the answer is a positive number. In this case, the area is 3/4. However, what if the interval for y=x^{3}+x was below the x-axis? Say, from -1 to 0? Wouldn't the answer then be a -3/4? If you use the Fundamental Theorem of Calculus, that is what you get, but to me, it doesn't make sense to have a "negative area."

Any thoughts?

Explanation / Answer

(0 to 1) x3 +x -0dx

=(0 to 1) (1/4)x4 +(1/2)x2 +c

=(1/4)14 +(1/2)12 +c -0-0-c

=3/4

(-1 to 0) 0-(x3 +x) dx

=(-1 to 0) -( (1/4)x4 +(1/2)x2 +c )

=-(0+0+c- (1/4)(-1)4 -(1/2)(-1)2 -c )

=3/4

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