Definite Integral

Definition

A definite integral is a mathematical concept that represents the signed area under a curve defined by a function over a specific interval on the x-axis. It provides a way to calculate the accumulation of quantities, such as area, volume, or displacement, over a given range and is closely related to the fundamental theorem of calculus, which connects differentiation and integration.

5 Must Know Facts For Your Next Test

1. The definite integral can be represented symbolically as $$\int_{a}^{b} f(x) \, dx$$, where $$a$$ and $$b$$ are the lower and upper limits of integration.
2. The result of a definite integral is a numerical value that represents the net area between the function and the x-axis from $$x=a$$ to $$x=b$$.
3. If the function is above the x-axis within the interval, the definite integral yields a positive value; if it is below, it results in a negative value.
4. The evaluation of a definite integral often involves finding an antiderivative using indefinite integration techniques and then applying the limits to get the final value.
5. The definite integral plays a crucial role in applications across various fields, including physics, engineering, and economics, by allowing for calculations of total quantities such as work done or total profit.

Review Questions

• How does the definite integral differ from the indefinite integral in terms of representation and application?
  • The definite integral differs from the indefinite integral primarily in its representation and application. While an indefinite integral represents a family of antiderivatives without specific bounds, symbolized as $$\int f(x) \, dx$$, a definite integral is expressed with limits and calculates the signed area under a curve between those limits, as in $$\int_{a}^{b} f(x) \, dx$$. The definite integral provides specific numerical results useful for applications like calculating total area or accumulated quantities.
• Explain how the Fundamental Theorem of Calculus connects differentiation and integration in evaluating definite integrals.
  • The Fundamental Theorem of Calculus establishes a vital connection between differentiation and integration by stating that if a function is continuous over an interval, then differentiating its integral gives back the original function. This theorem allows us to evaluate definite integrals by first finding an antiderivative of the function and then applying it at the upper and lower limits. The difference between these values yields the area under the curve over that interval, showcasing how these two operations are interrelated.
• Evaluate how Riemann sums can be used to approximate definite integrals and discuss their significance in understanding integration.
  • Riemann sums serve as foundational approximations for evaluating definite integrals by partitioning the interval into smaller subintervals and summing up areas of rectangles formed under the curve. By increasing the number of partitions and refining these sums through left, right, or midpoint evaluations, one approaches the exact value of the definite integral. This method highlights both conceptual understanding and practical techniques for integration while demonstrating how accumulating infinitesimally small quantities can yield precise total values.

Related terms

Indefinite Integral: An indefinite integral represents a family of functions and is expressed without specific limits, providing the antiderivative of a function.

Fundamental Theorem of Calculus: This theorem links the concept of differentiation with that of integration, stating that if a function is continuous on an interval, then the definite integral of its derivative over that interval equals the change in the function's values at the endpoints.

Riemann Sum: A Riemann sum is a method for approximating the value of a definite integral by dividing the area under the curve into small rectangles and summing their areas.