Triple Integral - Change the order of integration

sirhc1
Messages
4
Reaction score
0

Homework Statement



\int^{1}_{0}\int^{x^2}_{0}\int^{y}_{0} f(x,y,z) dz dy dx

Find 5 equivalent iterated integrals.

Homework Equations



0 ≤ z ≤ y

0 ≤ y ≤ x^2

0 ≤ x ≤ 1

The Attempt at a Solution



1) \int^{1}_{0}\int^{√y}_{0}\int^{x^2}_{0} f(x,y,z) dz dx dy

I will try dz dy dx first.

Because y = x^2, so 0 ≤ z ≤ x^2

Because y = x^2, so 0 ≤ x ≤ √y

And by the same logic, 0 ≤ y ≤ 1

When I integrate for f(x,y,z) = 1, the correct answer is 1/10. I do not get the same answer with my solution. Help! Is it possible to solve this without graphing it? Or is it necessary to get the correct answer?
 
Physics news on Phys.org
sirhc1 said:

Homework Statement



\int^{1}_{0}\int^{x^2}_{0}\int^{y}_{0} f(x,y,z) dz dy dx

Find 5 equivalent iterated integrals.

Homework Equations



0 ≤ z ≤ y

0 ≤ y ≤ x^2

0 ≤ x ≤ 1

The Attempt at a Solution



1) \int^{1}_{0}\int^{√y}_{0}\int^{x^2}_{0} f(x,y,z) dz dx dy

I will try dz dy dx first.

Because y = x^2, so 0 ≤ z ≤ x^2

Because y = x^2, so 0 ≤ x ≤ √y

Here is an error. For each x, y goes from 0 to x^2. If you graph that region in an xy-plane, it is below and to the right of the graph of y= x^2. That means that, for each y, x goes from \sqrt{y} up to 1. The y-integral is \int_{x^2}^1 dy

Looking at this, I realize now that I made that mistake in the previous post. I have edited it.

And by the same logic, 0 ≤ y ≤ 1

When I integrate for f(x,y,z) = 1, the correct answer is 1/10. I do not get the same answer with my solution. Help! Is it possible to solve this without graphing it? Or is it necessary to get the correct answer?
 
Last edited by a moderator:
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top