# Calculus 3 : Triple Integration in Cartesian Coordinates

## Example Questions

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### Example Question #131 : Multiple Integration

Evaluate the following integral

, where

Explanation:

### Example Question #132 : Multiple Integration

Find the value of the triple integral

Explanation:

An integral expression of the form

is equivalent to an integral of the form

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

It is worth noting that the order the integration is performed does not matter, since each variable will be integrated independently, with the others being treated as constants for the purposes of integration. For example

Considering our integral:

The approach is simply to take it step by step:

### Example Question #131 : Multiple Integration

Find the value of the triple integral

Explanation:

An integral expression of the form

is equivalent to an integral of the form

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

It is worth noting that the order the integration is performed does not matter, since each variable will be integrated independently, with the others being treated as constants for the purposes of integration. For example

Considering our integral:

The approach is simply to take it step by step:

### Example Question #134 : Multiple Integration

Find the value of the triple integral

Explanation:

An integral expression of the form

is equivalent to an integral of the form

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

It is worth noting that the order the integration is performed does not matter, since each variable will be integrated independently, with the others being treated as constants for the purposes of integration. For example

Considering our integral:

The approach is simply to take it step by step:

### Example Question #1 : Triple Integration In Cartesian Coordinates

Find the value of the triple integral

Explanation:

An integral expression of the form

is equivalent to an integral of the form

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

It is worth noting that the order the integration is performed does not matter, since each variable will be integrated independently, with the others being treated as constants for the purposes of integration. For example

Considering our integral:

The approach is simply to take it step by step. But first, we should consider the order in which we wish to integrate; those  and  terms in front of the exponent are daunting. Let's see if we can eliminate some by integrating with respect to  initially:

This is already looking more manageable. Now, however, we have both an  and  in front of our exponent, and we must decide which one to choose for the next integration. Recall that . Integrating with respect to  would be a better way of eliminating the terms in front of the exponent:

Finally, we're left with a (relatively) simpler integral:

Being careful about the order of integration can make the work much easier (relatively).

### Example Question #136 : Multiple Integration

Find the value of the triple integral

Explanation:

An integral expression of the form

is equivalent to an integral of the form

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

It is worth noting that the order the integration is performed does not matter, since each variable will be integrated independently, with the others being treated as constants for the purposes of integration. For example

Considering our integral:

The approach is simply to take it step by step; however, to make the job of integration easier, it might be prudent to begin by integrating with respect to :

That worked out pretty well in terms of getting rid of some terms in the front; let's use that trick again and take the integral with respect to  next:

This leaves a relatively straightforward integral with respect to :

Being careful about the order of integration can make a complicated problem a bit simpler.

### Example Question #137 : Multiple Integration

Find the value of the triple integral

Explanation:

An integral expression of the form

is equivalent to an integral of the form

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

It is worth noting that the order the integration is performed does not matter, since each variable will be integrated independently, with the others being treated as constants for the purposes of integration. For example

Considering our integral:

The approach is simply to take it step by step:

Explanation:

Explanation: