change order of integration and reparameterize to simplify a double integral

[serdyn321]

The limits of integration are {sqrt(y)<=x<=1, 0<=y<=1}

and the integrand is cos(x^3)

simply altering the order of integration does not simplify the problem. I have graphed the area of integration but i'm at a loss on how to reparameterize my limits of integration to simplify this problem.

I can compute once I have the new parameters.

anyone got any ideas?

 

[Deleted member 4993]

The limits of integration are {sqrt(y)<=x<=1, 0<=y<=1}

and the integrand is cos(x^3)

simply altering the order of integration does not simplify the problem. I have graphed the area of integration but i'm at a loss on how to reparameterize my limits of integration to simplify this problem.

I can compute once I have the new parameters.

anyone got any ideas?

.

You are sure that your integrand is cos(x^3) and NOT cos(x)^3 [ i.e. cos^3(x)]

The limits on y would be x^2 to 1.

 

[HallsofIvy]

The limits of integration are {sqrt(y)<=x<=1, 0<=y<=1}

and the integrand is cos(x^3)

simply altering the order of integration does not simplify the problem. I have graphed the area of integration but i'm at a loss on how to reparameterize my limits of integration to simplify this problem.

I can compute once I have the new parameters.

anyone got any ideas?

This is \(\displaystyle \int_0^1 \int_{\sqrt{y}}^1 cos(x^3)dx dy\)
The limits of integration say "y varies from 0 to 1 and, for each y, x varies from \(\displaystyle \sqrt{y}\) to 1".

It is useful to graph this. \(\displaystyle x= \sqrt{y}\), in the first quadrant, is \(\displaystyle y= x^2\) (Remember that \(\displaystyle \sqrt{y}\) indicates the positive number whose square is y). x can go from 0 to 1 and y, for each x, goes from 0 to \(\displaystyle x^2\). That is, the integral is
\(\displaystyle \int_0^1\int_0^{x^2} cos(x^3)dy dx\)

There is no "y" in the integrand so doing the first integral gives \(\displaystyle \int_0^1 x^2 cos(x^3)dx\) which is easy to integrate.
(The "parameterization" referred to is, I think, just the obvious substitution.)