Question

Integrating an integral is called ______ ______ .

Short answer

Integrating an integral is called double integration.

Step-by-step solution

Understanding Integral Notation

In calculus, an integral is a mathematical concept that represents the area under a curve in a graph. The integral can be indefinite or definite, involving functions and the processes to evaluate them. The notation for integration is usually shown as \( \int \), followed by a function, and then the differential \( dx \).

Different Types of Integrals

Two main types of integration are indefinite and definite integrals. Indefinite integration does not have upper and lower limits, while definite integration includes limits. Both types are fundamental operations in calculus.

Defining Double Integrals

When asked about integrating an integral, we are referring to the concept of a double integral. A double integral involves integrating a function over a two-dimensional area, expressed mathematically as \( \int \int f(x,y) \,dx \,dy \).

Contextualizing Double Integrals

Double integrals are used to compute volumes under surfaces in three-dimensional space and are an extension of the concept of an integral into two dimensions, similar to calculating areas but in a volumetric sense.

Key concepts

Integral Notation

Integral notation is a way to represent the concept of integration in a mathematical form. It typically uses the integral sign, which looks like a stretched letter 'S', and is represented by \( \int \). This notation is followed by the function being integrated, called the integrand, and a differential, such as \( dx \), indicating the variable of integration. So, when you see \( \int f(x) \, dx \), it means you're finding the integral of the function \( f(x) \) with respect to \( x \).

Integrals can either be indefinite or definite, and the integral notation varies slightly between the two. Understanding this notation is crucial, as it forms the backbone of solving integral problems in calculus.

• For indefinite integrals, a constant of integration, \( + C \), is added to account for any constant term that could have been part of the original function derived.
• Definite integrals use limits of integration, represented as numbers at the top and bottom of the integral sign, to define the bounds over which the function is being evaluated.

Indefinite Integrals

Indefinite integrals involve finding an antiderivative of a function. This means that you are retrieving a function that, when differentiated, gives back the original function. The result of an indefinite integral is not a single value but a family of functions, all differing by a constant. This is why we include \( + C \) in the result.

The process of finding indefinite integrals is also known as integration. One key thing to remember is that unlike definite integrals, indefinite integrals do not deal with specific limits. They provide a more general solution.

Here are some points to clarify the process:

• The integrand is the function you are integrating.
• The integral sign tells you to perform the integration operation.
• \( + C \) is added to show there could be a constant term as any derivative of a constant is zero.

Definite Integrals

Definite integrals give you a number as the result, representing the accumulated value over an interval or the area under the curve between two specified bounds. They differ from indefinite integrals as they compute a specific value rather than a function.

A definite integral is written with the integral sign and limits of integration, such as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the lower and upper bounds, respectively. The result of a definite integral is the net area between the graph of the function and the x-axis over the interval defined by \( a \) and \( b \).

Some important considerations for definite integrals:

• They can be used to calculate total distances traveled, areas, and other quantities that accumulate over an interval.
• If the area is below the x-axis, it is considered negative in terms of net area.
• Follows the Fundamental Theorem of Calculus, connecting derivatives and integrals.