Test practice: Antiderivatives, Area, & FTC

Front

What is the Fundamental Theorem of Calculus (FTC)?

Back

The Fundamental Theorem of Calculus links the concept of differentiation and integration, stating that if a function is continuous on [a, b], then the integral of its derivative over that interval is equal to the difference in the values of the function at the endpoints: \( \int_a^b f'(x)dx = f(b) - f(a) \.

Front

What is an antiderivative?

Back

An antiderivative of a function \( f(x) \) is a function \( F(x) \) such that \( F'(x) = f(x) \). It represents the reverse process of differentiation.

Front

How do you find the indefinite integral of a function?

Back

To find the indefinite integral of a function, you determine the antiderivative and add a constant of integration \( C \). For example, \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) for \( n \neq -1 \.

Front

What is the area under a curve?

Back

The area under a curve between two points can be found using definite integrals. It represents the accumulation of quantities and is calculated as \( \int_a^b f(x)dx \).

Front

What is u-substitution in integration?

Back

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

Front

How do you set up a definite integral to find the area between a curve and the x-axis?

Back

To set up a definite integral for the area between a curve \( f(x) \) and the x-axis over an interval \( [a, b] \), use \( \int_a^b |f(x)|dx \) to account for any portions below the x-axis.

Front

What is the integral of \( 15\sqrt[3]{x^2} \)?

Back

The integral of \( 15\sqrt[3]{x^2} \) is \( 9x^{\frac{5}{3}} + C \).

Front

What is the integral of \( \frac{1}{\sqrt[5]{x^4}} \)?

Back

The integral of \( \frac{1}{\sqrt[5]{x^4}} \) is \( 5x^{\frac{1}{5}} + C \).

Front

What is the process to evaluate a definite integral?

Back

To evaluate a definite integral, first find the antiderivative of the function, then apply the Fundamental Theorem of Calculus: \( F(b) - F(a) \), where \( F \) is the antiderivative.

Front

What does the constant of integration represent?

Back

The constant of integration \( C \) represents the family of all antiderivatives of a function, accounting for the fact that differentiation eliminates constant terms.