Find the area given the curves

[jwxie]

Homework Statement

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

Homework Equations

the usual integral fx - gx from a to b

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

 

[Mandark]

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.

 

[HallsofIvy]

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.