Definite Integrals

Definition

A definite integral is a mathematical operation that calculates the area under a curve on a graph between two specific points. It represents the accumulation of a quantity over an interval and is a fundamental concept in calculus that connects the ideas of differentiation and integration.

5 Must Know Facts For Your Next Test

1. Definite integrals are used to calculate the area under a curve, the total distance traveled, the average value of a function, and the net change of a quantity over an interval.
2. The definite integral of a function $f(x)$ over the interval $[a, b]$ is denoted by $\int_a^b f(x) \, dx$, where $a$ and $b$ are the lower and upper limits of integration, respectively.
3. The Fundamental Theorem of Calculus states that the definite integral of a function $f(x)$ over the interval $[a, b]$ is equal to the difference between the antiderivative of $f(x)$ evaluated at $b$ and the antiderivative of $f(x)$ evaluated at $a$.
4. Substitution, also known as the change of variable, is a technique used to evaluate definite integrals involving functions that can be expressed in terms of a new variable.
5. Definite integrals involving exponential, logarithmic, and trigonometric functions can be evaluated using the properties of these functions and the Fundamental Theorem of Calculus.

Review Questions

• Explain how the Fundamental Theorem of Calculus relates to the evaluation of definite integrals.
  • The Fundamental Theorem of Calculus establishes a direct connection between differentiation and integration, allowing definite integrals to be evaluated using antiderivatives. Specifically, the theorem states that the definite integral of a function $f(x)$ over the interval $[a, b]$ is equal to the difference between the antiderivative of $f(x)$ evaluated at $b$ and the antiderivative of $f(x)$ evaluated at $a$. This relationship provides a powerful tool for calculating definite integrals without the need for complex integration techniques.
• Describe how the method of substitution can be used to evaluate definite integrals involving functions that can be expressed in terms of a new variable.
  • The method of substitution, also known as the change of variable, is a technique used to evaluate definite integrals that can be expressed in terms of a new variable. The process involves identifying a suitable substitution, $u = g(x)$, where $g(x)$ is a function of the original variable $x$. This substitution allows the definite integral to be rewritten in terms of the new variable $u$, simplifying the integration and often leading to a more manageable form. The Fundamental Theorem of Calculus is then applied to evaluate the definite integral using the antiderivative of the new integrand.
• Analyze the role of definite integrals in the context of integrals involving exponential, logarithmic, and trigonometric functions, and explain how these functions can be used to evaluate such integrals.
  • Definite integrals involving exponential, logarithmic, and trigonometric functions are crucial in various applications of calculus. The properties of these functions, combined with the Fundamental Theorem of Calculus, allow for the efficient evaluation of such integrals. For example, the integral of an exponential function can be evaluated using the antiderivative of the exponential function, while integrals involving logarithmic functions can be evaluated using the properties of logarithms. Similarly, trigonometric integrals can be evaluated using the antiderivatives of trigonometric functions and the Fundamental Theorem of Calculus. Understanding the relationship between definite integrals and these specialized functions is essential for solving a wide range of calculus problems.

Related terms

Antiderivative: A function whose derivative is the original function, also known as the indefinite integral.

Riemann Sum: A method of approximating the area under a curve by dividing the interval into smaller subintervals and summing the areas of the rectangles formed by the subintervals.

Fundamental Theorem of Calculus: A theorem that establishes the relationship between differentiation and integration, allowing the definite integral to be evaluated using the antiderivative of the integrand.