Integral

Definition

An integral is a mathematical operation that calculates the area under a curve on a graph, representing the accumulation or summation of a quantity over an interval. It is a fundamental concept in calculus and is used to analyze and describe the relationship between a function and its rate of change.

5 Must Know Facts For Your Next Test

1. Integrals can be used to represent the accumulation of a quantity, such as distance, work, or charge, over an interval.
2. The integral sign, $\int$, denotes the integration operation, and the limits of integration specify the interval over which the integration is performed.
3. Integrals can be evaluated using various techniques, including substitution, integration by parts, and the use of tables of known integrals.
4. The Fundamental Theorem of Calculus states that the integral of a function is the antiderivative of that function, allowing for the evaluation of integrals using differentiation.
5. Integrals are essential in the analysis of motion, as they can be used to determine displacement from velocity or velocity from acceleration.

Review Questions

• Explain how integrals are used to represent the accumulation of a quantity over an interval.
  • Integrals are used to calculate the total accumulation of a quantity, such as distance, work, or charge, over a specified interval. The integral sign, $\int$, denotes the integration operation, and the limits of integration define the interval over which the integration is performed. By integrating a function, we are effectively summing up the infinitesimal contributions of the function over the given interval, allowing us to determine the overall accumulation of the quantity represented by the function.
• Describe the relationship between integrals and derivatives as established by the Fundamental Theorem of Calculus.
  • The Fundamental Theorem of Calculus states that the integral of a function is the antiderivative of that function. This means that the process of integration is the inverse of differentiation. If we have a function $f(x)$, we can find its integral $\int f(x) dx$ by determining the antiderivative of $f(x)$. Conversely, if we have an integral, we can evaluate it by finding the derivative of the integrand. This relationship between integrals and derivatives is a powerful tool in calculus, allowing for the evaluation of integrals using differentiation.
• Analyze how integrals are used in the analysis of motion, and explain how they can be used to determine displacement from velocity or velocity from acceleration.
  • Integrals play a crucial role in the analysis of motion, as they can be used to determine the relationship between different kinematic quantities. For example, if we have a function representing the velocity of an object, $v(t)$, we can integrate this function with respect to time to find the displacement of the object, $\int v(t) dt = s(t)$. Similarly, if we have a function representing the acceleration of an object, $a(t)$, we can integrate this function twice with respect to time to find the velocity, $\int a(t) dt = v(t)$, and then integrate the velocity function to find the displacement. This ability to move between different kinematic quantities using integrals is essential in the study of motion and the analysis of physical systems.

Related terms

Derivative:

The derivative is the rate of change of a function at a particular point, representing the instantaneous slope of the function's graph.

Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus establishes the relationship between integrals and derivatives, allowing for the evaluation of integrals using differentiation.

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