# Multiple Integral **The [[integral]] of a multivariable function.** ![[DoubleIntegral.svg|500]] *A double integral can be represented as the total signed volume of the region between the surface of a function and the plane on which the domain of integration is defined* A *multiple integral* is the extension of definite integrals to the integration of *functions of multiple variables*. A multiple integral is the *signed hypervolume* of the region bound by the function and the *domain of integration*. > [!NOTE]- Explanation > In the integral of single variable functions, the integral can be thought of as the limit of a sequence of [[Riemann Integral#Riemann sum\|Riemann sums]] which are the total area of a set of rectangular approximations of the integral. > > ![[RiemannSum.svg|500]] > *A Riemann sum of a single-variable function with the representation of a single term highlighted* > > For multivariable functions, the rectangular approximations are replaced with *hyperrectangular*, or generalisations of rectangles in higher dimensions, approximations instead. > > In double integrals, these approximations would be rectangular prisms and a Riemann sum would be the total volume of the approximations. > > ![[DoubleIntegralExplanation.svg|500]] > *A single term or approximation within a Riemann sum of a two-variable function; the whole integral is to taken over the domain of integration $D$* ## Notation The notation for a multiple integral is similar to that of a [[Integral#Notation\|definite integral]] but uses *nested integral signs*, with the first integral to be computed notated with the innermost sign. This notation is useful for when computing the multiple integral as an *iterated integral*. The domain of integration is either denoted with the *limits of each argument* on their corresponding integral signs or *abbreviated* on the innermost sign. If the domain of integration is denoted in abbreviated form, then a single integral sign may also be used. $\int_{a}^{b}\left[\int_{c}^{d}f(x,y)\;dy\right]\;dx=\iint_D f(x,y)\;dA=\int_{D} f(x,y)\;dA$ *Notation for a double integral* ## Evaluation ### Iterated integration The most common method of evaluating multiple integrals is to express them as *iterated integrals*. This reduces the multiple integral into a series of directly computable definite integrals with *respect to a single variable* while the others are treated as constant. #### Examples > [!NOTE]- Constant limits > Suppose the integral > $\int_{0}^{1}\int_{0}^{2}12x^{2}y^{3}\;dy\;dx$ > is to be determined. The inner integral with respect to $y$ is calculated first, with $x$ treated as a constant, then the outer integral is calculated. > $\begin{align*} > \int_{0}^{1}\int_{0}^{2}12x^{2}y^{3}\;dy\;dx&=\int_{0}^{1}\left[\frac{12x^{2}y^{4}}{4}\right]_{0}^{2}\;dx \\ > &=\int_{0}^{1}48x^{2}\;dx \\ > &=16 > \end{align*}$ > [!NOTE]- Functions as limits > Suppose the integral > $\iint_{D}x^{4}-2y\;dy\;dx$ > is to be determined over the domain $D=|y|\le x^{2}, |x|\le 1$. > $\begin{align*} > \iint_{D}x^{4}-2y\;dy\;dx&=\int_{-1}^{1}\int_{-x^2}^{x^2}x^{4}-2y\;dy\;dx\\ > &=\int_{-1}^{1}\left[x^{4}y-y^{2}\right]_{-x^{2}}^{x^2}\;dx\\ > &=\int_{-1}^{1}2x^{6}\;dx\\ > &=\frac{4}{7} > \end{align*}$ ### Changing the order of integration The *order of integration* of a multiple integral can be changed without its value changing. This is a common method to avoid more difficult integrals in a particular variable. > [!NOTE]- Example with constant limits > Suppose the integral > $\int_{0}^{1}\int_{0}^{2}12x^{2}y^{3}\;dy\;dx$ > is to be determined. For the sake of demonstration, the order of integration can be reversed then the integral can be calculated using iterated integrals. > $\begin{align*} > \int_{0}^{2}\int_{0}^{1}12x^{2}y^{3}\;dx\;dy &=\int _{0}^{2}\left[4x^{3}y^{3}\right]_{0}^{1}\;dy \\ > &=\int_{0}^{2}4y^{3}\;dy\\ > &=\left[y^{4}\right]_{0}^{2}\;dy\\ > &=16 > \end{align*}$ > This is the same result as shown previously.