Iterated Integrals

Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...

Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the constant...

Double integrals provide a practical method for determining the volume of liquid contained in tanks with irregularly shaped sides. To estimate the total volume, the base region of the tank is divided into many very small rectangular sections. Each section forms the base of a thin column of liquid, and the volumes of all these columns are added together across the entire region. This process produces an accurate representation of the total liquid volume inside the tank.The limits of integration...

Changing the order of integration can make a triple integral easier to evaluate without changing the solid region being measured. In this example, the solid is enclosed by a flat base, a slanted plane, two vertical planes, and a parabolic cylinder. The goal is to integrate ex over this three-dimensional region, so the main task is to describe the boundaries in an order that leads to the simplest calculation.One possible setup uses x as the innermost variable. In this arrangement, each line...

The Integral Test is a method for determining whether an infinite series converges or diverges by comparing the series to an improper integral. It is especially useful when the terms of a series are difficult to add directly, but they follow the values of a related continuous function. Instead of summing infinitely many discrete terms one by one, the test studies the area under a curve that represents the same pattern of decrease.A glow stick provides a useful example of this idea. After...