Definite Integrals

Definition

Definite integrals are a fundamental concept in calculus that represent the signed area under a curve between two specified points on the x-axis. This area can be interpreted both geometrically and analytically, serving to quantify the accumulation of quantities such as distance, area, or volume. Definite integrals are crucial for applications across various fields, providing a means to calculate exact values for continuous functions over a closed interval.

5 Must Know Facts For Your Next Test

1. Definite integrals are denoted as $$\int_{a}^{b} f(x) \, dx$$, where $$a$$ and $$b$$ are the lower and upper limits of integration respectively.
2. The result of a definite integral is a number that represents the net area between the function and the x-axis within the interval [$$a$$, $$b$$].
3. If the function is above the x-axis on the interval, the definite integral gives a positive value; if below, it gives a negative value.
4. Calculating a definite integral often involves finding an antiderivative of the function and then applying the limits of integration.
5. Definite integrals can be used in various applications, including physics for calculating work done or in economics for determining consumer surplus.

Review Questions

• How do definite integrals relate to Riemann sums in approximating areas under curves?
  • Definite integrals and Riemann sums are closely connected because Riemann sums serve as a method to approximate the area under a curve by summing up rectangular areas. As the number of rectangles increases and their width decreases (approaching zero), the Riemann sum approaches the value of the definite integral. This shows how definite integrals can be seen as the limit of Riemann sums as the partition becomes infinitely fine, providing an exact measure of the area under the curve.
• In what ways does the Fundamental Theorem of Calculus link differentiation and definite integrals?
  • The Fundamental Theorem of Calculus establishes a crucial connection between differentiation and definite integrals by stating that if a function is continuous on an interval, then its definite integral can be computed using its antiderivative. Specifically, it asserts that if $$F$$ is an antiderivative of $$f$$, then $$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$. This theorem not only justifies why we can find areas using antiderivatives but also highlights how integrating a function reverses differentiation.
• Evaluate how definite integrals can be applied in real-world scenarios, such as physics or economics.
  • Definite integrals play a significant role in real-world applications by providing precise measurements in various fields. In physics, for instance, they are used to calculate work done by a variable force over a distance by integrating the force function with respect to displacement. In economics, definite integrals help determine consumer surplus by integrating demand curves. These applications show how definite integrals can quantify accumulated quantities, leading to insights and solutions for practical problems.

Related terms

Indefinite Integrals: Indefinite integrals represent a family of functions whose derivative is the original function, typically including a constant of integration.

Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is continuous over an interval, the definite integral can be computed using its antiderivative.

Riemann Sums: Riemann sums are approximations of the definite integral using finite sums of function values at certain points multiplied by small intervals, leading to the concept of integration.