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Glossary

A

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 ∫03f(x)dx=5\int_{0}^{3} f(x) dx = 5 and ∫37f(x)dx=10\int_{3}^{7} f(x) dx = 10, the Additivity Property tells you that ∫07f(x)dx=15\int_{0}^{7} f(x) dx = 15.

C

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 ∫023exdx\int_{0}^{2} 3e^x dx, you can use the Constant Multiple Rule to write it as 3∫02exdx3 \int_{0}^{2} e^x dx.

D

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)v(t) from t=0t=0 to t=5t=5 seconds, you would calculate the definite integral ∫05v(t)dt\int_{0}^{5} v(t) dt.

I

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 ∫0πcos⁡(x)dx\int_{0}^{\pi} \cos(x) dx, the function cos⁡(x)\cos(x) is the integrand.

L

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 ∫27x2dx\int_{2}^{7} x^2 dx, the number 2 is the lower limit of integration.

R

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 ∫15f(x)dx=8\int_{1}^{5} f(x) dx = 8, the Reversal of Limits Property means ∫51f(x)dx=−8\int_{5}^{1} f(x) dx = -8.

S

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 ∫ab(x2+sin⁡x)dx\int_{a}^{b} (x^2 + \sin x) dx, the Sum/Difference Rule allows you to calculate ∫abx2dx+∫absin⁡xdx\int_{a}^{b} x^2 dx + \int_{a}^{b} \sin x dx separately.

U

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 ∫27x2dx\int_{2}^{7} x^2 dx, the number 7 is the upper limit of integration.

Z

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 ∫44(x3−2x)dx\int_{4}^{4} (x^3 - 2x) dx, the Zero Rule immediately tells you the answer is 0.