Lec 16: Double integrals | MIT 18.02 Multivariable Calculus, Fall 2007

The paragraph introduces the concept of double integrals as a natural progression from single-variable integrals to functions of two variables. It explains the motivation behind double integrals by relating them to the calculation of areas under a curve for single-variable functions and extends this to the calculation of volumes under surfaces for two-variable functions. The double integral is presented as a way to measure the volume under the graph of a function z = f(x, y) over a specified region in the xy-plane. The notation dA for a small area element is introduced, and the double integral is symbolized with two integral signs, indicating the integration over a region R.

This section delves into a more rigorous definition of double integrals by drawing an analogy to the Riemann sum approach used in single-variable calculus. The process involves dividing the region under consideration into small pieces, calculating the volume of a small box for each piece, and then summing these volumes to estimate the total volume. The double integral is defined as the limit of this process as the size of the pieces approaches zero. The paragraph also discusses the importance of choosing the right method for calculating double integrals, hinting at the reduction of a double integral to two single integrals, and the relevance of single-variable calculus techniques.

The paragraph explains the process of computing double integrals by reducing them to iterated integrals, which involve integrating with respect to one variable while treating the other as a constant. It describes the method of slicing the region by parallel planes and integrating over these slices to find the volume. The concept of an iterated integral is introduced, where the inner integral is performed first, followed by the outer integral. The importance of understanding the bounds of integration and how they may vary depending on the variable being integrated is emphasized.

This paragraph presents a practical example of calculating a double integral over a rectangular region. The function to be integrated is 1 - x^2 - y^2, and the region is defined by x between 0 and 1 and y between 0 and 1. The process involves setting up the iterated integral, computing the inner integral first with respect to y and then the outer integral with respect to x. The bounds for y are constant, simplifying the integration process. The final computation yields a volume of one third.

The paragraph discusses the importance of correctly identifying the bounds of integration for double integrals, especially when the region is not a simple rectangle. It explains how the bounds for the inner integral depend on the outer variable, using the example of integrating over a quarter disk defined by x^2 + y^2 = 1 for x between 0 and 1. The bounds for y are derived from the equation of the circle, showing that the range of y values changes with x.

This section provides a detailed example of integrating the same function, 1 - x^2 - y^2, but over a different region, specifically a quarter disk. The bounds for y are now dependent on x, with the upper limit given by the square root of (1 - x^2). The paragraph guides through the process of setting up the iterated integral, with the inner integral being with respect to y and the outer integral with respect to x, both ranging from 0 to 1.

The paragraph explores the concept of switching the order of integration for double integrals, emphasizing the need to carefully reconsider the bounds of integration when doing so. It contrasts simple cases where the region is a rectangle and the order can be switched without issue, with more complex cases requiring a deeper analysis of the region's geometry. The importance of understanding how the slices of the region align with the variables of integration is highlighted.

This paragraph presents a complex example where the order of integration is switched for an integral involving the function e^y/y with bounds y from x to sqrt(x) and x from 0 to 1. The process involves visualizing the region and determining the new bounds for x when integrating with respect to y first, and then y when integrating with respect to x first. The example demonstrates the complexity of changing the order of integration and the need for a clear understanding of the region's geometry.

The paragraph details the computation of the switched integral from the previous example. It shows how to perform the inner integral with respect to x and then the outer integral with respect to y. The computation involves recognizing the function as an exponential function and using integration techniques to find the antiderivative. The final result is obtained by evaluating the antiderivative at the bounds of integration.

The final paragraph previews the next topic to be covered: double integrals in polar coordinates. It suggests that the method of integrating in polar coordinates will provide an easier approach to certain types of double integrals, such as those involving circular or radial symmetry, which were difficult to handle with the current methods. The paragraph also hints at the applications of double integrals that will be explored in future lessons.