AB Integral Flashcard

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

Define the term 'definite integral'.

Back

A definite integral is an integral that has specified upper and lower limits, representing the net area under the curve of a function between those limits.

Front

What does the notation ∫ f(x) dx represent?

Back

The notation ∫ f(x) dx represents the indefinite integral of the function f(x), which is the family of all antiderivatives of f.

Front

Explain the concept of 'area under the curve' in relation to integrals.

Back

The area under the curve of a function f(x) from a to b is represented by the definite integral ∫_a^b f(x) dx, which calculates the total area between the curve and the x-axis over that interval.

Front

What is the difference between a definite integral and an indefinite integral?

Back

A definite integral has specific limits and results in a numerical value, while an indefinite integral does not have limits and results in a function plus a constant of integration.

Front

How do you evaluate the integral ∫ (3x^2) dx?

Back

The integral ∫ (3x^2) dx = x^3 + C, where C is the constant of integration.

Front

What is the power rule for integration?

Back

The power rule for integration states that ∫ x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1.

Front

Define 'antiderivative'.

Back

An antiderivative of a function f is a function F such that F' = f, meaning the derivative of F is f.

Front

What is the significance of the constant of integration (C)?

Back

The constant of integration (C) represents the family of all antiderivatives of a function, accounting for the fact that there are infinitely many functions that differ by a constant.

Front

Explain the concept of 'Riemann sums'.

Back

Riemann sums are a method for approximating the definite integral of a function by dividing the area under the curve into rectangles and summing their areas.