Du

Definition

In the context of integration, 'du' refers to the infinitesimal change in the independent variable, often denoted as 'dx' or 'dy'. It represents an infinitely small increment or differential of the independent variable, which is crucial in the application of integration techniques to find the total change or accumulation of a function over an interval.

5 Must Know Facts For Your Next Test

1. The 'du' in integration represents an infinitesimal change in the independent variable, which is crucial for applying integration techniques to find the total change or accumulation of a function over an interval.
2. Integration involves the summation of these infinitesimal changes (du) to find the total change in the dependent variable over the given interval.
3. The concept of 'du' is closely related to the idea of the derivative, as it represents the instantaneous rate of change of a function at a particular point.
4. The Riemann sum is a method of approximating the integral of a function by dividing the interval into smaller subintervals and summing the areas of the rectangles formed by the function values at the subinterval endpoints.
5. The use of 'du' in integration is essential for understanding and applying various integration techniques, such as substitution, integration by parts, and integration using trigonometric identities.

Review Questions

• Explain the role of 'du' in the context of integration and how it relates to the concept of the derivative.
  • The 'du' in integration represents an infinitesimal change in the independent variable, which is crucial for applying integration techniques to find the total change or accumulation of a function over an interval. Integration involves the summation of these infinitesimal changes (du) to find the total change in the dependent variable. The concept of 'du' is closely related to the idea of the derivative, as it represents the instantaneous rate of change of a function at a particular point. The use of 'du' in integration is essential for understanding and applying various integration techniques, such as substitution, integration by parts, and integration using trigonometric identities.
• Describe the relationship between 'du' and the Riemann sum in the context of integration.
  • The Riemann sum is a method of approximating the integral of a function by dividing the interval into smaller subintervals and summing the areas of the rectangles formed by the function values at the subinterval endpoints. The concept of 'du' is closely tied to the Riemann sum, as it represents the infinitesimal change in the independent variable used to construct the Riemann sum. By taking the limit of the Riemann sum as the subinterval size approaches zero, the sum of the infinitesimal changes (du) converges to the exact value of the integral, which is the total change or accumulation of the function over the given interval.
• Analyze the importance of understanding the concept of 'du' in the context of other integration techniques, such as substitution and integration by parts.
  • The understanding of the 'du' concept is crucial for applying various integration techniques, such as substitution and integration by parts. In substitution, the 'du' term is used to transform the original integral into a new integral with a different independent variable, allowing for simpler integration. Similarly, in integration by parts, the 'du' term is used to divide the integrand into two factors, one of which is differentiated, and the other is integrated, leading to a new integral that may be easier to evaluate. The ability to manipulate the 'du' term is essential for successfully applying these integration techniques and ultimately finding the total change or accumulation of a function over a given interval.

Related terms

Differential: A differential is an infinitesimal change in a variable, representing the instantaneous rate of change of a function at a particular point.

Integration:

Integration is the process of finding the total change or accumulation of a function over an interval by summing up the infinitesimal changes (du) of the independent variable.

Riemann Sum:

A Riemann sum is a method of approximating the integral of a function by dividing the interval into smaller subintervals and summing the areas of the rectangles formed by the function values at the subinterval endpoints.