# Find The Integral 6 /(3X – Integration By Parts (Formula And Walkthrough)

## Triple integrals can be evaluated in six different orders

There are six ways to express an iterated triple integral. While the function ???f(x,y,z)??? inside the integral always stays the same, the order of integration will change, and the limits of integration will change to match the order.

Đang xem: Integral 6

The only hard part of these problems is finding the limits of integration for each of the three individual integrals in each of the six triple iterated integrals.

Remember that, with all iterated integrals, you work your way from the inside toward the outside. So, if the integral ends in ???dx dy dz???, it means you integrate from the inside out, from left to right, first with respect to ???x???, then with respect to ???y???, and then with respect to ???z???.

???intintint_Ef(x,y,z) dx dy dz???

Since you’re integrating with respect to ???x??? first, and you’ll need to integrate with respect to ???y??? and ???z??? later, you need to have ???y??? and ???z??? variables left over after you integrate with respect to ???x??? and evaluate over the associated limits of integration. This means that the inner integral needs to have limits of integration in terms of ???y??? and ???z???.

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???intintint_{x(y,z)}^{x(y,z)}f(x,y,z) dx dy dz???

Once you’ve integrated with respect to ???x???, you’ll have only ???y??? and ???z??? variables remaining. You’ll integrate with respect to ???y???, plugging in the limits of integration associated with ???y???. Since you need to leave ???z??? variables in the function in order to later integrate with respect to ???z???, that means your limits of integration for ???y??? need to be in terms of ???z???.

???intint_{y(z)}^{y(z)}int_{x(y,z)}^{x(y,z)}f(x,y,z) dx dy dz???

Finally, once you’ve eliminated ???y??? and have only ???z??? variables remaining, the limits of integration associated with ???z??? should be constants, so that your final answer is a constant.

???int_{z}^{z}int_{y(z)}^{y(z)}int_{x(y,z)}^{x(y,z)}f(x,y,z) dx dy dz???

Notice in the chart below how all of the innermost integrals have limits of integration in terms of two variables, the second integral has limits of integration in terms of one variable, and the outermost integral has constant limits of integration.

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