What Are the Limits of Integration for a Double Integral in a Triangular Region?

[Tony11235]

Double integral of y^3, where D is the triangular region with vertices (0,0), (1,2), and (0,3). I can't figure out what the limits are.

D={(x,y)|0<=x<=3...is this even half way right?

 

[Maxos]

e. g. {0<x<1, 2x<y<-x+3}

 

[cronxeh]

Just add $$y^3$$ into the double integral and evaluate it. Keep in mind for simple structures like a triangle you can verify that your answer is correct simply but using trigonometric formulas for area.

In this case area of a triangle is= $$\frac{(b*h)}{2}=\frac{3*1}{2}$$

 

[HallsofIvy]

Double integral of y^3, where D is the triangular region with vertices (0,0), (1,2), and (0,3). I can't figure out what the limits are.

D={(x,y)|0<=x<=3...is this even half way right?

No, because the "3" in (0, 3) is a y-value not x. The first thing you should do is draw a picture- like cronxeh did. One side of the triangle is the line x= 0 (from (0,0) to (0,3)). It should be easy to see that the line from (0,0) to (1,2) is y= 2x (A line through (0,0) is always of the form y= mx and m= 2 makes y= 2(1)= 2. If you don't see that you can still calculate that the slope is $$\frac{2-0}{1-0}= 2$$. )
For the line from (0,3) to (1,2), you can argue that the equation of a line is always of the form y- y₀= m(x- x₀). $$m= \frac{2-3}{1-0}= -1$$. Taking (x₀,y₀) to be (0, 3), that gives
y- 3= 1(x-0) or y= x+ 3. If you take (x₀,y₀) to be (1, 2) instead, you get y- 2= 1(x- 1) which reduces to y= x+ 3 also.

Now, since x can range from 0 to 2, the "outer integral" will be $$\int_0^3 dx$$. For every x, y ranges from the lower of the two lines (the one from (0,0) to (1,2)), which is y= 2x, to the higher (the one from (0,3) to (1,2)), which is y= -x+ 3 or, if you prefer, y= 3- x. The "inner integral" will be $$\int_{y= 2x}^{3-x} dy$$.

Since the integrand is y³, the integral is
$$\int_{x=0}^2\int_{y= 2x}^{3-x} y^3 dy dx$$.

 

[HallsofIvy]

You could, by the way, do this "the other way around" (I wouldn't recommend it here- it's much more complicated. My point is to show how you can get the limits of integration.)

Overall, y ranges from 0 to 3. If we choose to use the y integral as the "outer integral", the limits would be from 0 to 3. For each y, the left sides is always x= 0 and so the lower limit for the "inner integral" is x= 0. However, there are two different lines that make up the right side: y= 2x and y= 3-x or, since we are using x as the variable for the inner integral and the limits of the inner integral cannot depend on x, x= (1/2)y and x= 3- y. Which of those we use as the upper depends on what y is. If y< 2, the right side is x= (1/2)y and if y> 2, the right side is x= 3- y. The simplest thing to do is to do that as two separate integrals and add them:
$$\int_{y=0}^2\int_{x=0}^{\frac{y}{2}}y^3 dxdy+ \int_{y=2}^3\int_{x=0}^{3-y}y^3dxdy$$.

 

[Tony11235]

Thanks. I really didn't think about it enough. :grumpy:

 

[okigetitnow]

is there a #1 answer rating?

 

[Rubik]

Why does x range from 0 to 2 and why does this give the outer integral from 0 to 3?

 

[SammyS]

Why does x range from 0 to 2 and why does this give the outer integral from 0 to 3?

No doubt, that was a typo (In post #4).

The limits of integration for x should be from 0 to 1.