Q. 51 - Calculus

Question

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

Exercise 41 with the order dy dx dz.

Step-by-Step Solution

Answer

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

Step 1. Given Information.

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

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

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

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