Final Calc 2 T/F

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

Back

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

Front

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

Back

The notation ∫ f(x) dx represents the integral of the function f(x) with respect to x, which can be either definite or indefinite.

Front

Explain the concept of convergence in improper integrals.

Back

Convergence in improper integrals refers to the condition where the integral approaches a finite limit as the variable approaches a certain value or infinity.

Front

What is the purpose of using u-substitution in integration?

Back

U-substitution is used to simplify the process of integration by substituting a part of the integrand with a new variable, making the integral easier to solve.

Front

State the integration by parts formula.

Back

The integration by parts formula is ∫ u dv = uv - ∫ v du, where u and v are differentiable functions.

Front

What is a Riemann sum?

Back

A Riemann sum is a method for approximating the total area under a curve by dividing it into rectangles and summing their areas.

Front

Define the term '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 area under a curve in calculus?

Back

The area under a curve represents the total accumulation of quantities, which can be interpreted in various contexts such as distance, probability, and total change.