Integration

Definition

Integration is a fundamental concept in calculus that describes the process of finding the area under a curve or the accumulation of a quantity over an interval. It is the inverse operation of differentiation, allowing us to determine the original function from its rate of change.

5 Must Know Facts For Your Next Test

1. Integration is used to find the area under a curve, the volume of a three-dimensional object, and the total change in a quantity over an interval.
2. The process of integration involves finding the antiderivative or indefinite integral of a function, which represents the accumulation of the function's rate of change.
3. Definite integrals are used to calculate the total change in a quantity over a specific interval, while indefinite integrals represent the general antiderivative of a function.
4. The Fundamental Theorem of Calculus states that the definite integral of a function is equal to the difference between the values of the antiderivative at the upper and lower bounds of the interval.
5. Integration techniques, such as substitution, integration by parts, and integration by trigonometric substitution, are used to evaluate more complex integrals.

Review Questions

• Explain how integration is used to find the area under a curve.
  • Integration is used to find the area under a curve by representing the area as the accumulation of the function's rate of change over a specific interval. The definite integral of a function over an interval gives the total area under the curve within that interval. This is based on the fact that the integral of a function is the antiderivative of that function, and the difference between the antiderivative values at the upper and lower bounds of the interval represents the total change in the function over that range, which corresponds to the area under the curve.
• Describe the relationship between differentiation and integration, as outlined by the Fundamental Theorem of Calculus.
  • The Fundamental Theorem of Calculus establishes the connection between differentiation and integration. It states that the definite integral of a function is equal to the difference between the values of the antiderivative (or indefinite integral) of that function at the upper and lower bounds of the interval. This means that integration is the inverse operation of differentiation, and the definite integral can be evaluated by finding the antiderivative of the function and subtracting the antiderivative values at the given endpoints. This theorem allows us to evaluate definite integrals without having to use the limit definition of the integral.
• Analyze how integration techniques, such as substitution and integration by parts, can be used to evaluate more complex integrals.
  • Integration techniques, such as substitution and integration by parts, are used to evaluate more complex integrals that cannot be easily solved using the basic integration rules. Substitution involves replacing the original variable with a new variable, which can simplify the integration process and allow the integral to be evaluated using the basic integration rules. Integration by parts, on the other hand, involves applying the product rule in reverse to split the integral into two parts, one of which can be integrated more easily. These techniques expand the range of integrals that can be evaluated, enabling the calculation of more complex areas under curves, volumes of three-dimensional objects, and total changes in quantities over intervals. The appropriate integration technique to use depends on the specific form of the integrand and the desired result.

Related terms

Definite Integral: A definite integral represents the accumulation of a quantity over a specific interval, providing the total change in the function over that range.

Indefinite Integral: An indefinite integral represents the general antiderivative of a function, without specifying the bounds of the interval.

Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus establishes the connection between differentiation and integration, allowing us to evaluate definite integrals using the original function.