Double integrals problem

**chella182** · May 5th 2009, 12:02 PM

Well, my problem isn't with the double integral bit, it's just with the integration once I've switched the limits et cetera. The original problem is:

$\int_{0}^{1}\int_{\sqrt{y}}^{1}2x^5y\cos{(xy^2)}dx dy$

...and I'm told to solve it by changing the order of integration. I get:

$\int_{0}^{1}\int_{0}^{x^2}2x^5y\cos{(xy^2)}dydx$

I know this forum's not here to check my work, but whether my new limits are correct or not, I still don't know how to integrate $2x^5y\cos{(xy^2)}$ with respect to y, taking x as a constant.

Can anyone help?

**TheEmptySet** · May 5th 2009, 12:11 PM

Originally Posted by chella182

Well, my problem isn't with the double integral bit, it's just with the integration once I've switched the limits et cetera. The original problem is:

$\int_{0}^{1}\int_{\sqrt{y}}^{1}2x^5y\cos{(xy^2)}dx dy$

...and I'm told to solve it by changing the order of integration. I get:

$\int_{0}^{1}\int_{0}^{x^2}2x^5y\cos{(xy^2)}dydx$

I know this forum's not here to check my work, but whether my new limits are correct or not, I still don't know how to integrate $2x^5y\cos{(xy^2)}$ with respect to y, taking x as a constant.

Can anyone help?

I didn't check your change of order of inegration but use this sub to evaluate the integral

$u=xy^2 \implies du=2xydy$

and we get

$\int_{0}^{1}\int_{0}^{x^2}2x^5y\cos{(xy^2)}dydx$

$\int_{0}^{1}\int x^4\cos(u)dudx$

**redsoxfan325** · May 5th 2009, 12:15 PM

Originally Posted by chella182

Well, my problem isn't with the double integral bit, it's just with the integration once I've switched the limits et cetera. The original problem is:

$\int_{0}^{1}\int_{\sqrt{y}}^{1}2x^5y\cos{(xy^2)}dx dy$

...and I'm told to solve it by changing the order of integration. I get:

$\int_{0}^{1}\int_{0}^{x^2}2x^5y\cos{(xy^2)}dydx$

I know this forum's not here to check my work, but whether my new limits are correct or not, I still don't know how to integrate $2x^5y\cos{(xy^2)}$ with respect to y, taking x as a constant.

Can anyone help?

Your reordering of the integration is correct. Proceeding from there, let $u=y^2$ . Thus $du = 2y\,dy$

You now have $\int_0^1\int_0^{x^2}x^5\cos(xu)\,du = \frac{1}{x}\cdot x^5\sin(xu) = \left[x^4\sin(xy^2)\right]_0^{x^2} = x^4\sin(x^5)$

So we have $\int_0^1 x^4\sin(x^5)\,dx$

Let $u=x^5$ and $du=5x^4$ . Now you have:

$\frac{1}{5}\int_0^1\sin(u)\,du = -\frac{1}{5}\cos(u) = \left[-\frac{1}{5}\cos(x^5)\right]_0^1 = \boxed{\frac{1}{5}-\frac{1}{5}\cos(1)}$

**chella182** · May 5th 2009, 12:34 PM

I totally don't understand the second line of your working, sorry

where does the $2y$ disappear to? And the $x^4$ in the second integral? And I don't get where the $\frac{1}{x}$ has come from

**Pinkk** · May 5th 2009, 12:36 PM

$2y$ is the derivative of $y^{2}$ . All he did was a u-substitution.

**chella182** · May 5th 2009, 12:37 PM

No, I know that, where does it disappear to though?

**Pinkk** · May 5th 2009, 12:41 PM

Into the $du$ . It never "disappears", it's just part of his new variables $u$ and $du$ . And the $\frac{1}{x}$ is part of the integration of $cos(xy)$ .

The antiderivative of $cos(xy)$ where $x$ is treated as a variable is $\frac{sin(xy)}{x}$ .

**chella182** · May 5th 2009, 12:42 PM

So because the $2y$ appears in the $du$ it's not in the problem anymore? :S I'm well confused.

I get where the $\frac{1}{x}$ comes from now though, cheers.

**Pinkk** · May 5th 2009, 12:44 PM

It's just u-substitution.

Just replace the $y^{2}$ with a $u$ and $2ydy$ with a $du$ . Don't let the fact that it's a double integral throw you off. It's just a straightforward u-substitution.

**chella182** · May 5th 2009, 12:48 PM

I know I just... don't get it :S urgh. I'll go see my lecturer tomorrow if I get chance.

**Pinkk** · May 5th 2009, 12:58 PM

Let's look at a single integral:

$\int 2ycos(y^{2})\,\,dy$

Let $u=y^{2}$ and $du=2ydy$ .

Now just substitute:

$\int cos(u)\,\,du$

Now just integrate:

$\int cos(u)\,\,du = sin(u)$

Now just resubstitute for $u=y^{2}$ :

$sin(y^{2})$ .

Take the derivative to clarify:

$\frac{d}{dy}sin(y^{2})=cos(y^{2})\cdot \frac{d}{dy}y^{2} = 2ycos(y^{2})$

All you're doing is substituting one type of variable for another. The actual function isn't changed, just the way it's being represented.

**chella182** · May 5th 2009, 01:03 PM

Ohhh right, I get it. All you needed to do to explain was say $dy=\frac{du}{2y}$ so the $2y$ 's cancel

**Pinkk** · May 5th 2009, 01:05 PM

It hasn't disappeared. $du=2ydy$

I'm just representing $2ydy$ as $du$ so I can have an easier integral to evaluate.

And yes, you can think of it that way if you want. Whatever clicks with you.

**chella182** · May 5th 2009, 01:08 PM

It has cancelled out therefore it's disappeared in my mind. I do understand the whole thing, all I didn't get was why that disappeared 'cause no one was telling me where it'd gone. Although this does seem rather more complicated than examples we've done in lectures :S

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