Question

In Exercises 45–52, rewrite the indicated integral with the specified order of integration.

Exercise 41 with the order dy dx dz.

Short answer

The given indicated integral with the specified order of integration is${\int }_{-3}^3{\int }_{-\sqrt{9-z^2}}^{\sqrt{9-z^2}}{\int }_{-\sqrt{9-x^2-z^2}}^{\sqrt{9-x^2-z^2}}f(x,y,z)dydxdz.$

Step-by-step solution

Step 1. Given Information.

The given triple integral is${\int }_{-3}^3{\int }_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}{\int }_{-\sqrt{9-x^2-y^2}}^{\sqrt{9-x^2-y^2}}f(x,y,z)dzdydx.$

Step 2. Rewrite the integral.

We have to write the given integral in order of dy dx dz.

So, the limit of the integration will vary as follows:

The limit of y alters from $-\sqrt{9-x^2-z^2}\text{to}\sqrt{9-x^2-z^2}.$

The limit of xalters from $-\sqrt{9-z^2}\text{to}\sqrt{9-z^2}\text{.}$

The limit of zalters from $-3\text{to}3.$

Hence the indicated integral with the specified order of integration is${\int }_{-3}^3{\int }_{-\sqrt{9-z^2}}^{\sqrt{9-z^2}}{\int }_{-\sqrt{9-x^2-z^2}}^{\sqrt{9-x^2-z^2}}f(x,y,z)dydxdz.$