Change the order of integration and Evaluate21 20xxxydydx

To change the order of integration, we need to rewrite the given integral in terms of the other variable.

The given integral is ∫∫21 20xxxydydx.

To change the order of integration, we first need to determine the limits of integration for the new variables.

The original limits of integration for x are from 2 to 1, and for y are from 0 to x.

To change the order of integration, we need to express the limits of integration for the new variables.

The new limits of integration for y will be from 0 to 1, and for x will be from y to 2.

Now, we can rewrite the integral with the new order of integration:

∫∫21 20xxxydydx = ∫∫10 21 20xxxydxdy.

To evaluate this integral, we can integrate with respect to x first, and then with respect to y.

Integrating with respect to x, we get:

∫∫10 21 20xxxydxdy = ∫10 20xy2|y=0 to y=1 dy.

Simplifying this, we have:

∫10 20y2 dy = 20∫10 y2 dy.

Integrating with respect to y, we get:

20∫10 y2 dy = 20[y3/3]|y=0 to y=1.

Evaluating this, we have:

20[y3/3]|y=0 to y=1 = 20[(1)3/3 - (0)3/3] = 20(1/3) = 20/3.

Therefore, the value of the integral ∫∫21 20xxxydydx, with the order of integration changed, is 20/3.