1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    i2u0q8.jpg

    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

    HallsofIvy

    User Avatar
    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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook