Set up the integrals that can be used to find the area of the region by integrating with respect to X and Y
(the region is bounded by y=3x,y=x,y=4-x)
Anyone help me set up the integrals??
Did you make a sketch?
Here is Wolfram's take on the situation.
http://www.wolframalpha.com/input/?i=plot+y%3D3x%2Cy%3Dx%2Cy%3D4-x
You need the intersection of the lines
I assume you want the area of the enclosed region.
the intersection points are (0,0), (1,3), and (2,2)
from x = 0 to 1, y = 3x is above y = 3x
so the effective height = 3x-x = 2x
from 1 to 2, y = 4-x is above y = x
so the effective height is 4-x - x = 4 - 2x
So the area
= ∫2x dx from 0 to 1 + ∫(4-2x) dx from 1 to 2
take over from here
To find the area of the region bounded by the given curves, we need to set up two integrals: one with respect to x and another with respect to y.
First, let's determine the x-limits of integration. These limits will be determined by the intersection points of the curves.
1. The line y = 3x intersects y = x when 3x = x. Solving this equation gives x = 1.
2. The line y = 3x intersects y = 4 - x when 3x = 4 - x. Solving this equation gives x = 4/4 = 1.
So, the x-limits of integration are from x = 1 to x = 1.
Next, let's determine the y-limits of integration. These limits will be determined by the intersection points of the curves.
1. The line y = 3x intersects y = x when 3x = x. Solving this equation gives y = 3(1) = 3.
2. The line y = 3x intersects y = 4 - x when 3x = 4 - x. Solving this equation gives y = 4 - (4/3) = 8/3.
So, the y-limits of integration are from y = 3 to y = 8/3.
Now, we can set up the integrals:
1. Integrating with respect to x:
Since we are integrating with respect to x, the integrand should be in terms of x.
The height of the region varies with x, so the integrand will be (top curve - bottom curve) with respect to x.
∫[1 to 1] (3x - x) dx
2. Integrating with respect to y:
Since we are integrating with respect to y, the integrand should be in terms of y.
The width of the region varies with y, so the integrand will be (right curve - left curve) with respect to y.
∫[3 to 8/3] (4 - y) - y/3 dy
These are the two integrals that can be used to find the area of the region. Note that you will need to evaluate these integrals to obtain the actual area.