Define the integrand

Multiple integrals extend summation from one dimension to two or higher dimensions. The key is not to compute first—but to clearly sketch the region of integration.

I always begin by drawing the region and labeling its upper and lower bounds. If the region is incorrect, even the neatest computation is meaningless.

In the previous article, we explored multivariable functions and their partial derivatives—laying the essential groundwork for understanding how to compute multiple integrals. Multiple integrals are a central concept in multivariable calculus, used to compute the “volume” or “total quantity” over a region in multidimensional space. In this article, we’ll introduce the concept of multiple integrals in detail, walk through systematic computational methods, and reinforce understanding with concrete examples.

Fundamental Concepts of Multiple Integrals

A multiple integral is the process of integrating a multivariable function over a specified region. Specifically, given a function $f(x, y)$, we aim to compute its double integral over region $D$:

$$\iint_D f(x, y) \, dx \, dy$$

This expression means summing (in the limit sense) the values of $f(x, y)$ across all points $(x, y)$ in $D$, yielding an overall “volume” or “aggregate quantity.”

Describing Region $D$

Region $D$ may be as simple as a rectangle—or as complex as a curved, irregular domain. Typically, we approximate $D$ by subdividing it into small rectangles, evaluate $f$ on each subrectangle, sum those values, and then take the limit as subdivisions become infinitely fine—yielding the formal definition of the double integral.

Step-by-Step Procedure for Computing Multiple Integrals

Computing a multiple integral generally follows these steps:

1. Identify the region of integration $D$: Clearly define the shape and extent of the domain.
2. Choose the order of integration: Decide whether to integrate with respect to $x$ first and then $y$, or vice versa.
3. Set the integration limits: Determine the bounds for each variable based on the geometry of $D$.
4. Evaluate the inner integral: Compute the integral with respect to the inner variable.
5. Evaluate the outer integral: Integrate the result of the inner integral with respect to the outer variable.

We now illustrate these steps with a concrete example.

Practical Example: Computing a Double Integral

Suppose we have the function $f(x, y) = x^2 + y^2$, and wish to compute its double integral over the rectangular region $D = [0, 1] \times [0, 1]$.

Step 1: Identify Region $D$

Here, $D$ is the unit square in the Cartesian plane: $0 \leq x \leq 1$ and $0 \leq y \leq 1$.

Step 2: Choose Integration Order

We choose to integrate with respect to $x$ first, then $y$.

When computing multiple integrals:

• First, sketch the region of integration;
• Then determine the integration order;
• Next, write down clear upper and lower limits;
• Finally, verify consistency between the integrand and its geometric interpretation.

Step 3: Set Integration Limits

Thus, the double integral becomes:

$$\iint_D f(x, y) \, dx \, dy = \int_0^1 \int_0^1 (x^2 + y^2) \, dx \, dy$$

Step 4: Evaluate the Inner Integral

Compute the inner integral with respect to $x$:

$$\int_0^1 (x^2 + y^2) \, dx = \int_0^1 x^2 \, dx + \int_0^1 y^2 \, dx$$

First term:

$$\int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3}$$

Since $y^2$ is constant with respect to $x$:

$$\int_0^1 y^2 \, dx = y^2 \cdot \int_0^1 1 \, dx = y^2$$

So the inner integral evaluates to:

$$\int_0^1 (x^2 + y^2) \, dx = \frac{1}{3} + y^2$$

Step 5: Evaluate the Outer Integral

Substitute the result into the outer integral:

After reading Multiple Integrals in Multivariable Calculus, don’t stop at “I understand.” Go back, pick one step, and work through it yourself—then note where you get stuck. This makes future learning more solid.

$$\int_0^1 \left( \frac{1}{3} + y^2 \right) \, dy = \int_0^1 \frac{1}{3} \, dy + \int_0^1 y^2 \, dy$$

First term:

$$\int_0^1 \frac{1}{3} \, dy = \frac{1}{3} \cdot \left[ y \right]_0^1 = \frac{1}{3}$$

Second term:

$$\int_0^1 y^2 \, dy = \left[ \frac{y^3}{3} \right]_0^1 = \frac{1}{3}$$

Therefore:

$$\int_0^1 \left( \frac{1}{3} + y^2 \right) \, dy = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$$

Hence, the value of the double integral is:

$$\iint_D f(x, y) \, dx \, dy = \frac{2}{3}$$

Python Implementation

We can compute this double integral numerically using scipy.integrate.dblquad. Below is the corresponding code:

from scipy.integrate import dblquad

# Define the integrand
def integrand(x, y):
    return x**2 + y**2

# Define integration bounds
x_lower = 0
x_upper = 1
y_lower = 0
y_upper = 1

# Compute the double integral
result, error = dblquad(integrand, x_lower, x_upper, lambda x: y_lower, lambda x: y_upper)

print(f"Double integral result: {result:.2f}, Estimated error: {error:.2e}")

Running this code confirms our manual calculation.

When reviewing Multiple Integrals in Multivariable Calculus, place key concepts, procedural steps, and observable outcomes side-by-side on a single page for efficient recall.

When practicing Multiple Integrals in Multivariable Calculus, write input conditions, computational actions, and resulting outputs together—making future review faster and more reliable.

Summary

In this article, we thoroughly examined the concept and computation of multiple integrals, supported by a worked example to deepen intuition about multivariable calculus. In upcoming articles, we’ll explore real-world applications of multivariable calculus—broadening our conceptual toolkit further.