Double and triple integrals

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Describe the region first. In the inner integral, treat the outer variable as constant. Integrate from inside out and apply bounds at every stage. Changing the order requires describing the same region again; merely swapping dx and dy is insufficient. Fubini applies in regular cases, subject to integrability conditions.

Worked example

1. Identify the integrand, variables and order. For multiple integrals, start with the innermost integral.

   $$\int\limits_{0}^{1}\int\limits_{0}^{1 - y} \left(x + y\right)\, dx\, dy$$

2. Integrate with respect to x. Treat the other variables as constants at this stage.

   $$\int \left(x + y\right)\, dx$$

3. Split the sum. The integral of a sum is the sum of its integrals.

   $$\int \left(x + y\right)\, dx = \int x\, dx + \int y\, dx$$

4. Apply the power rule: add 1 to the exponent and divide by the new exponent. This requires n ≠ −1.

   $$\int x\, dx = \frac{x^{2}}{2}$$

5. Integrate a constant by multiplying it by the variable.

   $$\int y\, dx = x y$$

6. Combine the results of the substeps and simplify.

   $$\int \left(x + y\right)\, dx = \frac{x^{2}}{2} + x y$$

7. Check this primitive by differentiating it: recover exactly the integrand of this stage.

   $$x + y = x + y$$

8. Substitute the upper bound first and then the lower bound. Keep parentheses when subtracting.

   $$F(1 - y)-F(0) = y \left(1 - y\right) + \frac{\left(1 - y\right)^{2}}{2} - \left(0\right)$$

9. Apply the bounds: upper value minus lower value. For singularities or infinite bounds, the engine evaluates the integral as improper.

   $$\left[\frac{x^{2}}{2} + x y\right]_{0}^{1 - y} = y \left(1 - y\right) + \frac{\left(1 - y\right)^{2}}{2}$$

10. Integrate with respect to y. Treat the other variables as constants at this stage.

    $$\int \left(y \left(1 - y\right) + \frac{\left(1 - y\right)^{2}}{2}\right)\, dy$$

11. Split the sum. The integral of a sum is the sum of its integrals.

    $$\int \left(y \left(1 - y\right) + \frac{\left(1 - y\right)^{2}}{2}\right)\, dy = \int \frac{\left(1 - y\right)^{2}}{2}\, dy + \int y \left(1 - y\right)\, dy$$

12. Make a substitution and replace the differential too; both must change together.

    $$u = - y,\quad du = -1\,dy$$

13. Split the sum. The integral of a sum is the sum of its integrals.

    $$\int \left(u^{2} + u\right)\, du = \int u^{2}\, du + \int u\, du$$

14. Apply the power rule: add 1 to the exponent and divide by the new exponent. This requires n ≠ −1.

    $$\int u^{2}\, du = \frac{u^{3}}{3}$$

15. Apply the power rule: add 1 to the exponent and divide by the new exponent. This requires n ≠ −1.

    $$\int u\, du = \frac{u^{2}}{2}$$

16. Combine the results of the substeps and simplify.

    $$\int \left(u^{2} + u\right)\, du = \frac{u^{3}}{3} + \frac{u^{2}}{2}$$

17. After integrating the expression in u, return to the original variable.

    $$\int y \left(1 - y\right)\, dy = - \frac{y^{3}}{3} + \frac{y^{2}}{2}$$

18. Move the constant factor outside: it does not depend on the integration variable.

    $$\int \frac{\left(1 - y\right)^{2}}{2}\, dy = - \frac{\left(1 - y\right)^{3}}{6}$$

19. Make a substitution and replace the differential too; both must change together.

    $$u = 1 - y,\quad du = -1\,dy$$

20. Move the constant factor outside: it does not depend on the integration variable.

    $$\int \left(- u^{2}\right)\, du = - \frac{u^{3}}{3}$$

21. Apply the power rule: add 1 to the exponent and divide by the new exponent. This requires n ≠ −1.

    $$\int u^{2}\, du = \frac{u^{3}}{3}$$

22. Combine the results of the substeps and simplify.

    $$\int \left(- u^{2}\right)\, du = - \frac{u^{3}}{3}$$

23. After integrating the expression in u, return to the original variable.

    $$\int \left(1 - y\right)^{2}\, dy = - \frac{\left(1 - y\right)^{3}}{3}$$

24. Combine the results of the substeps and simplify.

    $$\int \frac{\left(1 - y\right)^{2}}{2}\, dy = - \frac{\left(1 - y\right)^{3}}{6}$$

25. Combine the results of the substeps and simplify.

    $$\int \left(y \left(1 - y\right) + \frac{\left(1 - y\right)^{2}}{2}\right)\, dy = - \frac{y^{3}}{3} + \frac{y^{2}}{2} - \frac{\left(1 - y\right)^{3}}{6}$$

26. Check this primitive by differentiating it: recover exactly the integrand of this stage.

    $$- y^{2} + y + \frac{\left(1 - y\right)^{2}}{2} = y \left(1 - y\right) + \frac{\left(1 - y\right)^{2}}{2}$$

27. Substitute the upper bound first and then the lower bound. Keep parentheses when subtracting.

    $$F(1)-F(0) = \frac{1}{6} - \left(- \frac{1}{6}\right)$$

28. Apply the bounds: upper value minus lower value. For singularities or infinite bounds, the engine evaluates the integral as improper.

    $$\left[- \frac{y^{3}}{3} + \frac{y^{2}}{2} - \frac{\left(1 - y\right)^{3}}{6}\right]_{0}^{1} = \frac{1}{3}$$

29. The result corresponds to the specified bounds and integration order.

    $$\frac{1}{3}$$