a double integral question.........?

Question

hi guys...can you help me with my homeowrk, i've tried so many resources at home related to my homework but i can't figure out how to solve this one....

here it goes......

use the double integral to find the volume of a solid bounded by the coordinate planes and the surface x^1/2 + y^1/2 + z^1/2 = a^1/2........

but in the condition that it would be with dx and dy in it for the solutions...

can you help me out with this one?.....

thanks a lot in advance..........

have a nice day.....

Answer 1

What this question means is that they want an integral of the form:
Volume = Integral (Integral (Something * dy) * dx)

First, you have to decide what that Something is. Integrals are really sums (this is the really basic tenet of calculus that it's easy to lose sight of) of very small pieces. In this case you are trying to find a volume so you need three dimensions.
Specifically, you are summing (over) dx * dy (on the xy plane), in other words, small little tiles. To get a volume, imagine that those small little tiles are acutally the ends of bricks, and the length of the brick is z. If you add up the volume of all those bricks, then you'll get the volume under the original curve. That is to say, Something = z.

So far, this has covered the general case of finding volume. Now we'll see how to apply this to z = (√a - √x - √y)²
Along the x axis, x goes all the way from 0 to a.
For each x, y goes from 0 to (√a-√x)²
We got that last part by setting z to 0 since we're on the xy plane.

The end is straightforward because we just have a simple (though messy) integral to calculate:
The inner integral is (√a - √x - √y)² dy from 0 to (√a-√x)²:
That is:
Integral(a+x+y-2√a√x - 2√a√y + 2√x√y dy) =
ay + xy + y²/2 - 2y√a√x - 4y√y(√a+√x)/3 evaluated from 0 to (√a-√x)²
The lower bound gives 0, so this is just:
(a+x)(√a-√x)² + (√a-√x)^4/2 -2√a√x(√a-√x)² - 4(√a-√x)³(√a+√x)/3 =
(a+x-4/3(a-x))(√a-√x)² + (√a-√x)^4/2 -2√a√x(√a-√x)² =
2(a/3+5x/3)(√a-√x)² - 3√a√x(√a-√x)² =
(a/6+17x/6 - 3√a√x)(√a-√x)²

Now you should integrate this last expression (times dx) from x = 0 to x = a and if there are no calculational errors that will be your volume. Remember that a is a constant and you are integrating with respect to x, just as in the above integral we integrated with respect to y and treated both a and x as constant.