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Find the area given the curves

  1. Feb 16, 2010 #1
    1. The problem statement, all variables and given/known data

    sketch and find the area of the region
    bounded by the given curves. Choose the variable of integration
    so that the area is written as a single integral

    y = x, y = 2, y = 6 − x, y = 0

    2. Relevant equations

    the usual integral fx - gx from a to b

    3. The attempt at a solution

    here is my graph. my book gave an answer inte [0,1] (6-2y)dy = 8


    how come?

    okay, so according to my graph above, i don't know where to start. unlike y = x and y = x^2 and y = 0 they have a definite intersection

    i find this problem difficult because all these curves intersect at different points
  2. jcsd
  3. Feb 17, 2010 #2
    Looks like the region they're after is the trapezium. You'll want to integrate with respect to y so that the area can be expressed as a single integral.
  4. Feb 17, 2010 #3


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    Science Advisor

    The region bounded by those lines has four vertices. They are
    1) The vertex where y= x and y= 0 intersect: (0, 0)
    2) The vertex where y= x and y= 2 intersect: (2, 2)
    3) The vertex where y= 2 and y= 6- x intersect: 2= 6-x so x= 4, (4, 2)
    4) The vertex where y= 6- x and y= 0 intersect: (6, 0).

    As Mandark says, the simplest integral is to integrate with y going from 0 to 2, the two horizontal lines. The integrand will be (6- x)- 1= 6- 2x.

    The hard way would be to integrate with respect to x. You would have to divide it into 3 integrals: first integrate from x= 0 to x= 2 with integrand x- 0, then from x= 2 to x= 4, with integrand 2- 0, then from x= 4 to x= 6 with integrand 6- 2x- 0.

    Of course, this is, as Mandark also says, a trapezoid (trapezium) with bases of lengths 6 and 2 and height 2. The simplest way to find the area is to use the formula for area of a trapezoid.
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