Double & Triple Integral Solver | Multivariable Calculus Tool

DOUBLE & TRIPLE INTEGRAL SOLVER

Iterated Multivariable Integration Over General Regions

Multivariable Calculus

Integrand Function $f(x, y)$:

Outer Bound $x_1$ (a):

Outer Bound $x_2$ (b):

Middle Bound $y_1(x)$:

Middle Bound $y_2(x)$:

Iterated Integral Results

Integral Value ($I$) 0.3333

Step-by-Step Derivation & Bounds Setup

Double & Triple Integral Solver

In multivariable calculus, evaluating double integrals ($\iint_R f(x,y) \,dA$) and triple integrals ($\iiint_V f(x,y,z) \,dV$) involves setting up iterated integrals with variable limits to calculate surface areas, bounded volumes, and total mass distribution.

1. Fubini's Theorem & Region Formulations

Integral Type	Region Description	Iterated Integral Setup
Double (Type I)	$a \le x \le b, \, g_1(x) \le y \le g_2(x)$	$$\int_{a}^{b} \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \, dx$$
Double (Type II)	$c \le y \le d, \, h_1(y) \le x \le h_2(y)$	$$\int_{c}^{d} \int_{h_1(y)}^{h_2(y)} f(x,y) \, dx \, dy$$
Triple Integral	$a \le x \le b, \, g_1(x) \le y \le g_2(x), \, h_1(x,y) \le z \le h_2(x,y)$	$$\int_{a}^{b} \int_{g_1(x)}^{g_2(x)} \int_{h_1(x,y)}^{h_2(x,y)} f(x,y,z) \, dz \, dy \, dx$$

2. Applications of Multivariable Integration

Volume Calculation: Setting $f(x,y) = 1$ gives the total area of $R$. Setting $f(x,y,z) = 1$ yields the total 3D volume of region $V$.
Center of Mass & Centroids: Integrating density functions $\rho(x,y,z)$ computes total mass $M$ and coordinate moments ($M_{yz}, M_{xz}, M_{xy}$).

Frequently Asked Questions

What is the order of integration?

The order of differentials ($dz \, dy \, dx$) specifies which inner variable is evaluated first. Limits for inner integrals can depend on outer variables, but the outermost bounds must always be constant numbers.

Can bounds contain variable functions?

Yes. Inner bounds $y_1(x), y_2(x)$ or $z_1(x,y), z_2(x,y)$ accommodate non-rectangular 2D and 3D geometric regions.

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Double & Triple Integral Solver | Multivariable Calculus Tool