Iterated Integrals - setting up limits of integration

[mirandasatterley]

Homework Statement

Find the volume of the region under the graph of f(x,y) = x+y and above the region y²≤x, 0≤x≤9

The Attempt at a Solution

From these equations, x will be integrated from 0-9, but I'm not sure about y.

My thinking is that y will be intgrated from 0-3 because y²≤x and the smallest value of x is 0, and the square root of 0 is 0, so that is the smallest y, and the largest x value possible is 9 and the positive quare root of 9 is 3, so this is the largest value of y,
So I would integrate:

∫0-9∫0-3 (x+y)dydx.

Is this correct or am I missing something, where the limits of integration also involve using f(x,y)?

 

[HallsofIvy]

Homework Statement

Find the volume of the region under the graph of f(x,y) = x+y and above the region y²≤x, 0≤x≤9

The Attempt at a Solution

From these equations, x will be integrated from 0-9, but I'm not sure about y.

My thinking is that y will be intgrated from 0-3 because y²≤x and the smallest value of x is 0, and the square root of 0 is 0, so that is the smallest y, and the largest x value possible is 9 and the positive quare root of 9 is 3, so this is the largest value of y,
So I would integrate:

∫0-9∫0-3 (x+y)dydx.

Is this correct or am I missing something, where the limits of integration also involve using f(x,y)?

No, that is not correct. The region $a\le x\le b$, $c\le y\le d$, with a, b, c, d numbers is always a rectangle and the figure here is not a rectangle. But the bounds do NOT involve f(x,y)- that is a "z" value and goes inside the integral as you have it.

Always draw a picture for problems like this. y^2= x is a parabola, of course, "on its side". The line x= 9 is a vertical line crossing the parabola at (9,3) and at (9, -3). Yes, you can integrate with x going from 0 to 9. On your picture, mark an arbitrary "x" by marking a point on the x-axis between 0 and 9. Now draw a vertical line from one boundary to the other. The y bounds, for that x, are y values of those endpoints: $(x, -\sqrt{x})$, and $(x, -\sqrt{x})$. Your integral is
$$\int{x=0}^9\int_{y=-\sqrt{x}}^{\sqrt{x}} f(x)dy dx= \int{x=0}^9\int_{y=-\sqrt{x}}^{\sqrt{x}}(x+ y) dy dx$$

Of course, like any double integral, you can reverse the order of integration. If you look at your picture you will see that y ranges, overall, from -3 to 3. Draw a horizontal line across the parabola, representing an arbitrary value of y in that range. It will have left endpoint on the parabola: x= y^2, and right endpoint on the vertical line x= 9. Those will now be the limits of integration for this order:
$$\int_{y= -3}^3\int_{x= y^2}^9 f(x,y)dx dy= \int_{y= -3}^3\int_{x= y^2}^9(x+ y)dx dy$$

Try it both ways. You should get the same answer.

 

[mirandasatterley]

In a similar situation where i have to switch the order of integration from
∫0-3∫y²-9 f(x,y) dxdy to dydx,

Is this also a parabola on it's side, with intercepts through (3,9) and (0,9), meaning that I would be setting up the new limits of integration, fro the portion of the parabola above y=0?

 

[mirandasatterley]

So, ∫0-9∫Square root of x -3 f(x,y) dydx