Flashcard Basic Integration and the 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

How do you find the area under a curve defined by a polynomial function?

Back

To find the area under a curve defined by a polynomial function, you need to compute the definite integral of the function over the given interval. For example, for \( f(x) = 5x^4 + 3x + 7 \) from 0 to 4, calculate \( \int_0^4 (5x^4 + 3x + 7)dx \.

Front

Evaluate the integral \( \int_1^5 (-x^2 + 6x - 10)dx \).

Back

The result of the integral is \( -\frac{28}{3} \).

Front

What is the integral of \( e^{5x} \)?

Back

The integral of \( e^{5x} \) is \( \frac{1}{5}e^{5x} + c \).

Front

What is the integral of \( \frac{1}{2x} \)?

Back

The integral of \( \frac{1}{2x} \) is \( \frac{1}{2} \ln |2x| + c \).

Front

What is a definite integral?

Back

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

Front

What is an indefinite integral?

Back

An indefinite integral represents a family of functions and includes a constant of integration (c). It does not have specified limits.

Front

What is the power rule for integration?

Back

The power rule for integration states that \( \int x^n dx = \frac{x^{n+1}}{n+1} + c \) for any real number \( n \neq -1 \).

Front

How do you evaluate \( \int (5x^4 + 3x + 7)dx \)?

Back

Using the power rule, \( \int (5x^4 + 3x + 7)dx = x^5 + \frac{3}{2}x^2 + 7x + c \).

Front

What is the area under the curve of a function?

Back

The area under the curve of a function between two points is found by calculating the definite integral of the function over that interval.