Definite Integrals

Front

What is a definite integral?

Back

A definite integral is a mathematical concept that represents the signed area under a curve defined by a function, between two specified limits (a and b). It is denoted as ∫[a,b] f(x) dx.

Front

What is the Fundamental Theorem of Calculus?

Back

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then ∫[a,b] f(x) dx = F(b) - F(a).

Front

How do you evaluate a definite integral using u-substitution?

Back

To evaluate a definite integral using u-substitution, you first choose a substitution u = g(x) that simplifies the integral. Then, change the limits of integration according to the substitution and replace dx with du/g'(x). Finally, evaluate the integral in terms of u and substitute back to x.

Front

What is the geometric interpretation of a definite integral?

Back

The geometric interpretation of a definite integral is the area under the curve of the function f(x) from x = a to x = b, taking into account the sign of the area (above or below the x-axis).

Front

What is the difference between definite and indefinite integrals?

Back

A definite integral has specific limits of integration and results in a numerical value representing area, while an indefinite integral does not have limits and represents a family of functions (antiderivatives) plus a constant of integration.

Front

What is the role of limits in definite integrals?

Back

Limits in definite integrals define the interval over which the area under the curve is calculated. They specify the starting point (a) and the endpoint (b) of the integration.

Front

What is the notation for a definite integral?

Back

The notation for a definite integral is ∫[a,b] f(x) dx, where f(x) is the integrand, a is the lower limit, b is the upper limit, and dx indicates the variable of integration.

Front

How can definite integrals be used in real-world applications?

Back

Definite integrals can be used in various real-world applications such as calculating areas, volumes, total accumulated quantities, and solving problems in physics, engineering, and economics.

Front

What is the trapezoidal rule in the context of definite integrals?

Back

The trapezoidal rule is a numerical method for estimating the value of a definite integral by dividing the area under the curve into trapezoids and summing their areas.

Front

What is the relationship between the area under a curve and the definite integral?

Back

The area under a curve from x = a to x = b is equal to the value of the definite integral ∫[a,b] f(x) dx, which calculates the net area considering the function's position relative to the x-axis.