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Math Help - area between 2 curves using |f(x)dx-g(x)dx|

  1. #1
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    area between 2 curves using |f(x)dx-g(x)dx|

    problem:






    Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width.



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    Here is what the 2 lines look like:

    In doing these types of area problems when part of the curve is negative, would i set up the integral to be from -1 to 0 of 6x*dx-(x^2-7)*dx and add it to the integral of 0 to 7 of 6x*dx-(x^2-7)*dx?

    thanks.
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  2. #2
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    First, if y= |6x| then your graph is wrong. You have y= 6x.

    Because both |6x| and x^2- 7 are even functions, the graphs are symmetric about the y-axis and I would recommend just integrating on one side then doubling. For x> 0, |6x|= 6x so the two graphs will cross, on the right, when x^2- 7= 6x which is the same as x^2- 6x- 7= (x- 7)(x+ 1)= 0. The only positive root of that is x= 7 so the area is 2\int_0^7 6x- (x^2- 7) dx.

    If it really is y= 6x rather than y= |6x|, just integrate \int_{-1}^7 6x- (x^2- 7)dx. You don't need to worry about changing signs because 6x is always larger than x^2- 7 on that interval so 6x- (x^2- 7) is always positive.
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  3. #3
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    thank you, i spotted my error.
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