Lecture #25: Double and Iterated Integrals
MATH 145 – Calculus I
November 6, 2025
Integrating \(f(x, y)\)
How do we integrate a function \(f(x, y)\)?
How is the integral \(\displaystyle\int_a^b f(x) \, dx\) defined and evaluated?
Can we extend the ideas of Riemann sums to multivariable functions?
Approximating the Integral of \(f(x, y)\)
In the one variable case, we took intervals of the number line and estimated the area under the curve as the sum of rectangles.
In the two variable case, let’s take intervals of both axes and use the rectangles to make rectangular prisms to estimate the volume under the surface.
You can expand this idea to \(n\)-dimensional functions, but we won’t touch on that.
Approximating the Integral of \(f(x, y)\)
Defining the Integral of \(f(x, y)\)
Let \(f\) be a continuous function on a rectangle \(R\).
A double Riemann sum for \(f\) over the rectangle \(R\) is
\[ \sum_{j=1}^n \sum_{i=1}^m \left( f(x_{ij}^*, y_{ij}^*) \times \delta A \right). \]
If \(f\) is continuous over \(R\), then the double integral of \(f\) over \(R\) is
\[ \iint_R f(x, y) \, dA = \lim\limits_{m,n \to \infty} \sum_{i=1}^m \left( f(x_{ij}^*, y_{ij}^*) \times \delta A \right). \]
Properties of the Double Integral
Let \(f\) and \(g\) be continuous functions on a rectangle \(R = \{ (x, y) : a \le x \le b, c \le x \le d \}\), and let \(k\) be a constant.
Then:
- \(\displaystyle\iint_R \left( f(x, y) + g(x, y) \right) \, dA = \displaystyle\iint_R f(x, y) \, dA + \displaystyle\iint_R g(x, y) \, dA\).
- \(\displaystyle\iint_R k f(x, y) \, dA = k \displaystyle\iint_R f(x, y) \, dA\).
- If \(f(x, y) \ge g(x, y)\) on \(R\), then \(\displaystyle\iint_R f(x, y) \, dA \ge \displaystyle\iint_R g(x, y) \, dA\).
Evaluating Double Integrals
If \(f = f(x, y)\) is a continuous function on a rectangle \(R = [a, b] \times [c, d]\), then
\[ \iint_R f(x, y) \, dA = \int_c^d \int_a^b f(x, y) \, dx \, dy = \int_a^b \int_c^d f(x, y) \, dy \, dx. \]
Example: Evaluating Double Integrals
Example: Find the volume under the graph of \(f(x, y) = x^2 + y^2\) over the rectangle \(-2 \le x \le 2\), \(-2 \le y \le 2\).
Example: Evaluating Double Integrals
Example: Let \(f(x, y) = 25 - x^2 - y^2\) on the rectangular domain \(R = [-3, 3] \times [-4, 4]\). Find \(\displaystyle\iint_R f(x, y) \, dA\).
Example: Find \(\displaystyle\iint_R \sin(x) \cos(y) \, dA\) for \(R = \left\{ (x, y) : 0 \le x \le \frac{\pi}{3}, 0 \le y \le \frac{\pi}{4} \right\}\).
