Glossary

Additivity Property

Criticality: 3

A property that allows a definite integral over a larger interval to be expressed as the sum of definite integrals over adjacent sub-intervals.

Example:

If you know $\int_{0}^{3} f(x) dx = 5$ and $\int_{3}^{7} f(x) dx = 10$ , the Additivity Property tells you that $\int_{0}^{7} f(x) dx = 15$ .

Constant Multiple Rule

Criticality: 3

A property of definite integrals that permits a constant factor within the integrand to be moved outside the integral sign.

Example:

To simplify $\int_{0}^{2} 3e^x dx$ , you can use the Constant Multiple Rule to write it as $3 \int_{0}^{2} e^x dx$ .

Definite Integral

Criticality: 3

An integral with specified upper and lower limits, which evaluates to a numerical value representing the net signed area under the curve of a function over a given interval.

Example:

To find the total displacement of a car whose velocity is given by $v(t)$ from $t=0$ to $t=5$ seconds, you would calculate the definite integral $\int_{0}^{5} v(t) dt$ .

Integrand

Criticality: 2

The function that is being integrated within a definite or indefinite integral, typically denoted as $f(x)$ in $\int f(x) dx$.

Example:

For the integral $\int_{0}^{\pi} \cos(x) dx$ , the function $\cos(x)$ is the integrand.

Lower Limit of Integration

Criticality: 3

The smaller number or variable placed at the bottom of the integral symbol, indicating the starting point of the interval over which the integration is performed.

Example:

In the expression $\int_{2}^{7} x^2 dx$ , the number 2 is the lower limit of integration.

Reversal of Limits Property

Criticality: 3

A property that allows the swapping of the upper and lower limits of integration by negating the entire definite integral.

Example:

Given that $\int_{1}^{5} f(x) dx = 8$ , the Reversal of Limits Property means $\int_{5}^{1} f(x) dx = -8$ .

Sum/Difference Rule

Criticality: 3

A property stating that the definite integral of a sum or difference of functions can be evaluated as the sum or difference of their individual definite integrals.

Example:

If you need to integrate $\int_{a}^{b} (x^2 + \sin x) dx$ , the Sum/Difference Rule allows you to calculate $\int_{a}^{b} x^2 dx + \int_{a}^{b} \sin x dx$ separately.

Upper Limit of Integration

Criticality: 3

The larger number or variable placed at the top of the integral symbol, indicating the endpoint of the interval over which the integration is performed.

Example:

In the expression $\int_{2}^{7} x^2 dx$ , the number 7 is the upper limit of integration.

Zero Rule

Criticality: 2

A property of definite integrals stating that if the upper and lower limits of integration are identical, the value of the integral is zero.

Example:

If you need to evaluate $\int_{4}^{4} (x^3 - 2x) dx$ , the Zero Rule immediately tells you the answer is 0.

Glossary

Additivity Property

Criticality: 3

A property that allows a definite integral over a larger interval to be expressed as the sum of definite integrals over adjacent sub-intervals.

Example:

If you know $\int_{0}^{3} f(x) dx = 5$ and $\int_{3}^{7} f(x) dx = 10$ , the Additivity Property tells you that $\int_{0}^{7} f(x) dx = 15$ .

Constant Multiple Rule

Criticality: 3

A property of definite integrals that permits a constant factor within the integrand to be moved outside the integral sign.

Example:

To simplify $\int_{0}^{2} 3e^x dx$ , you can use the Constant Multiple Rule to write it as $3 \int_{0}^{2} e^x dx$ .

Definite Integral

Criticality: 3

An integral with specified upper and lower limits, which evaluates to a numerical value representing the net signed area under the curve of a function over a given interval.

Example:

To find the total displacement of a car whose velocity is given by $v(t)$ from $t=0$ to $t=5$ seconds, you would calculate the definite integral $\int_{0}^{5} v(t) dt$ .

Integrand

Criticality: 2

The function that is being integrated within a definite or indefinite integral, typically denoted as $f(x)$ in $\int f(x) dx$.

Example:

For the integral $\int_{0}^{\pi} \cos(x) dx$ , the function $\cos(x)$ is the integrand.

Lower Limit of Integration

Criticality: 3

The smaller number or variable placed at the bottom of the integral symbol, indicating the starting point of the interval over which the integration is performed.

Example:

In the expression $\int_{2}^{7} x^2 dx$ , the number 2 is the lower limit of integration.

Reversal of Limits Property

Criticality: 3

A property that allows the swapping of the upper and lower limits of integration by negating the entire definite integral.

Example:

Given that $\int_{1}^{5} f(x) dx = 8$ , the Reversal of Limits Property means $\int_{5}^{1} f(x) dx = -8$ .

Sum/Difference Rule

Criticality: 3

A property stating that the definite integral of a sum or difference of functions can be evaluated as the sum or difference of their individual definite integrals.

Example:

If you need to integrate $\int_{a}^{b} (x^2 + \sin x) dx$ , the Sum/Difference Rule allows you to calculate $\int_{a}^{b} x^2 dx + \int_{a}^{b} \sin x dx$ separately.

Upper Limit of Integration

Criticality: 3

The larger number or variable placed at the top of the integral symbol, indicating the endpoint of the interval over which the integration is performed.

Example:

In the expression $\int_{2}^{7} x^2 dx$ , the number 7 is the upper limit of integration.

Zero Rule

Criticality: 2

A property of definite integrals stating that if the upper and lower limits of integration are identical, the value of the integral is zero.

Example:

If you need to evaluate $\int_{4}^{4} (x^3 - 2x) dx$ , the Zero Rule immediately tells you the answer is 0.