Glossary

Absolute Value

Criticality: 3

The non-negative value of a number or expression, used in area calculations to ensure all contributions are positive regardless of whether the curve is above or below the x-axis.

Example:

When calculating the total distance a particle travels, even if it moves backward, you take the absolute value of its displacement for each segment to sum them up.

Absolute Value

Criticality: 2

The non-negative magnitude of a real number, representing its distance from zero regardless of its sign.

Example:

If a definite integral yields $-4.5$ , the actual area is the absolute value of $-4.5$ , which is $4.5$ square units.

Absolute Value (Modulus)

Criticality: 3

The non-negative magnitude of a real number, representing its distance from zero regardless of its sign.

Example:

When calculating the actual physical area, you must take the absolute value of the definite integral, as area cannot be negative.

Accumulation of Change

Criticality: 2

The conceptual understanding that a definite integral sums up infinitesimal changes of a quantity over an interval to find the total change or net effect.

Example:

If a function represents the rate of water flowing into a tank, the accumulation of change over an hour tells you the total volume of water added.

Area Between Two Curves

Criticality: 3

The region enclosed by two functions over a specified interval, calculated using definite integrals.

Example:

To find the Area Between Two Curves like y=x² and y=x+2, you integrate the difference of the functions over their intersection points.

Area Between a Curve and the Y-Axis

Criticality: 3

The region bounded by a curve, the y-axis, and two horizontal lines (or y-intercepts), calculated using integration with respect to y.

Example:

Calculating the area between a curve and the y-axis for $x = y^2$ from $y=0$ to $y=2$ helps visualize the space enclosed.

Area Between a Curve and the x-Axis

Criticality: 3

The region bounded by a function's graph, the x-axis, and two vertical lines (limits of integration), calculated using definite integrals.

Example:

To determine the total ground covered by a projectile's parabolic path, you would calculate the area between a curve and the x-axis.

Areas Partly Above and Below the x-axis

Criticality: 3

A method for calculating the total area enclosed by a curve and the x-axis when parts of the curve are above and parts are below the x-axis, requiring separate integration for each segment and summing absolute values.

Example:

To find the total area of the region bounded by y = cos(x) and the x-axis from x=0 to 2π, you must use the areas partly above and below the x-axis method, integrating separately for positive and negative regions.

Bounds (Limits of Integration)

Criticality: 3

The upper and lower values of the independent variable that define the specific interval over which a definite integral is evaluated.

Example:

In ∫₀¹ (x²) dx, 0 and 1 are the Bounds or Limits of Integration.

Curve

Criticality: 1

A continuous line or outline, typically representing the graph of a function in a coordinate plane.

Example:

The path of a thrown ball can be modeled by a parabolic curve, and calculus helps find the area under it.

Curves

Criticality: 1

Graphical representations of functions, typically plotted on a coordinate plane, whose shapes are defined by their equations.

Example:

The parabola y=x² and the line y=x are examples of Curves whose enclosed area can be calculated.

Definite Integral

Criticality: 3

The integral of a function over a specified interval, representing the net area between the function and the x-axis.

Example:

To calculate the total displacement of a rocket from launch (t=0) to engine cutoff (t=10s) given its velocity function, you would compute the definite integral of the velocity over that time interval.

Definite Integral

Criticality: 3

An integral that computes the net accumulation of a quantity, such as the area under a curve or between curves, over a specific interval defined by upper and lower limits.

Example:

Calculating the Definite Integral of a velocity function from t=0 to t=5 gives the total displacement over that time.

Definite Integral

Criticality: 3

A mathematical operation that calculates the net signed area between a function's graph and the x-axis over a specified interval.

Example:

Using a definite integral from 0 to 5 for a velocity function $v(t)$ gives the total displacement of an object over that time.

Definite Integral

Criticality: 3

An integral with specified upper and lower limits, used to calculate the net area under a curve or between curves over a given interval.

Example:

Using a definite integral like $\int_{1}^{3} x^2 , dx$ to find the exact area under the parabola from $x=1$ to $x=3$ .

Function Order (Above/Below Function)

Criticality: 3

The principle of identifying which function has greater y-values (or x-values for integration with respect to y) over a given interval, ensuring the integrand is non-negative.

Example:

When integrating with respect to x, always place the Function Order of the curve that is graphically above the other first in the integrand to ensure a positive area.

Function x = g(y)

Criticality: 3

A mathematical relationship where the independent variable is $y$ and the dependent variable is $x$, meaning $x$ is expressed in terms of $y$.

Example:

To find the area with respect to the y-axis, you must first rearrange $y = x^2 - 4$ into the form of a function x = g(y), which would be $x = \pm\sqrt{y+4}$ .

Integrand

Criticality: 2

The function that is being integrated within an integral expression.

Example:

In the integral ∫(x² + 3x) dx, the expression (x² + 3x) is the Integrand.

Integration Limits

Criticality: 3

The upper and lower bounds of a definite integral, which define the specific interval over which the integration is performed.

Example:

When finding the area between $x=y^2$ and the y-axis from $y=1$ to $y=3$ , $y=1$ and $y=3$ are the integration limits.

Integration with respect to x

Criticality: 3

The process of finding the area between curves by setting up the integral in terms of the variable x, typically when functions are given as y = f(x).

Example:

To find the area between y=x² and y=x+2, you would use Integration with respect to x, setting up the integral as ∫(x+2 - x²) dx.

Integration with respect to y

Criticality: 3

The process of finding the area between curves by setting up the integral in terms of the variable y, typically when functions are given as x = f(y).

Example:

If you have curves like x=y² and x=y+2, you would use Integration with respect to y, setting up the integral as ∫(y+2 - y²) dy.

Intersection Points

Criticality: 3

Points where two curves meet or cross each other, which are critical for determining the limits of integration when calculating areas between curves.

Example:

To find the area enclosed by the parabola y = x^2 and the line y = x + 2, you first identify their intersection points to define the boundaries of integration.

Intersection Points

Criticality: 3

The specific coordinates where two or more curves meet or cross each other, often serving as the limits of integration when finding the area between them.

Example:

Finding the Intersection Points of y=x² and y=4 is the first step to determine the bounds for calculating the area enclosed by them.

Interval

Criticality: 2

A continuous range of values for a variable, often denoted by [a, b], over which a function is considered or an integral is evaluated.

Example:

When finding the area between two curves from x=0 to x=2, the Interval is [0, 2].

Limits (of integration)

Criticality: 3

The upper and lower bounds of an interval over which a definite integral is calculated, defining the specific region of interest.

Example:

When finding the area under $y=x^2$ from $x=1$ to $x=3$ , 1 and 3 are the limits of integration.

Logarithms

Criticality: 2

The inverse operation to exponentiation, used to find the power to which a base number must be raised to produce a given number.

Example:

When solving for x in 2^x = 8, you use Logarithms to find that x = log₂(8) = 3.

Multiple Integrals

Criticality: 2

In this context, it refers to the strategy of splitting a complex area calculation into several definite integrals, often due to changes in which function is 'on top' or where the curve crosses the x-axis.

Example:

When finding the area of a region bounded by three different functions, you might need to use multiple integrals, each with different limits and integrand functions, to cover the entire region.

Negative Area Integrals

Criticality: 2

A definite integral whose value is negative, indicating that the area lies predominantly to the left of the y-axis (or below the x-axis for $\int f(x) dx$).

Example:

If you integrate $x = y^2 - 9$ from $y=0$ to $y=2$ , the result will be a negative area integral because the curve is to the left of the y-axis in that interval.

Negative Areas

Criticality: 3

A result of a definite integral when the function's graph lies below the x-axis over the interval, indicating a signed area rather than a physical area.

Example:

If a definite integral for a profit function yields a negative area, it signifies a net loss over that period.

Top Function

Criticality: 2

In the context of finding the area between two curves, the function with the higher y-value (or further to the right for integration with respect to y) within a specified interval.

Example:

When finding the area between y = √x and y = x^2 from x=0 to x=1, y = √x is the top function because its values are greater than or equal to y = x^2 in that interval.

Total Areas Between Two Curves

Criticality: 3

A method for calculating the total area enclosed by two intersecting curves, typically involving integrating the difference between the 'top' and 'bottom' functions over relevant intervals.

Example:

Determining the land area shared by two overlapping circular fields would involve calculating the total areas between two curves representing the field boundaries.

Y-Axis Intercepts

Criticality: 2

The points where a graph crosses or touches the y-axis, occurring when the x-coordinate is zero.

Example:

For the curve $x = y^2 - 4$ , the y-axis intercepts are found by setting $x=0$ , yielding $y=\pm 2$ .

x-axis intercepts

Criticality: 2

The points where a function's graph crosses or touches the x-axis, meaning the y-coordinate is zero.

Example:

Finding the x-axis intercepts of a quadratic function helps determine the starting and ending points for calculating the area it encloses with the x-axis.

Glossary

Absolute Value

Criticality: 3

The non-negative value of a number or expression, used in area calculations to ensure all contributions are positive regardless of whether the curve is above or below the x-axis.

Example:

When calculating the total distance a particle travels, even if it moves backward, you take the absolute value of its displacement for each segment to sum them up.

Absolute Value

Criticality: 2

The non-negative magnitude of a real number, representing its distance from zero regardless of its sign.

Example:

If a definite integral yields $-4.5$ , the actual area is the absolute value of $-4.5$ , which is $4.5$ square units.

Absolute Value (Modulus)

Criticality: 3

The non-negative magnitude of a real number, representing its distance from zero regardless of its sign.

Example:

When calculating the actual physical area, you must take the absolute value of the definite integral, as area cannot be negative.

Accumulation of Change

Criticality: 2

The conceptual understanding that a definite integral sums up infinitesimal changes of a quantity over an interval to find the total change or net effect.

Example:

If a function represents the rate of water flowing into a tank, the accumulation of change over an hour tells you the total volume of water added.

Area Between Two Curves

Criticality: 3

The region enclosed by two functions over a specified interval, calculated using definite integrals.

Example:

To find the Area Between Two Curves like y=x² and y=x+2, you integrate the difference of the functions over their intersection points.

Area Between a Curve and the Y-Axis

Criticality: 3

The region bounded by a curve, the y-axis, and two horizontal lines (or y-intercepts), calculated using integration with respect to y.

Example:

Calculating the area between a curve and the y-axis for $x = y^2$ from $y=0$ to $y=2$ helps visualize the space enclosed.

Area Between a Curve and the x-Axis

Criticality: 3

The region bounded by a function's graph, the x-axis, and two vertical lines (limits of integration), calculated using definite integrals.

Example:

To determine the total ground covered by a projectile's parabolic path, you would calculate the area between a curve and the x-axis.

Areas Partly Above and Below the x-axis

Criticality: 3

Example:

Bounds (Limits of Integration)

Criticality: 3

The upper and lower values of the independent variable that define the specific interval over which a definite integral is evaluated.

Example:

In ∫₀¹ (x²) dx, 0 and 1 are the Bounds or Limits of Integration.

Curve

Criticality: 1

A continuous line or outline, typically representing the graph of a function in a coordinate plane.

Example:

The path of a thrown ball can be modeled by a parabolic curve, and calculus helps find the area under it.

Curves

Criticality: 1

Graphical representations of functions, typically plotted on a coordinate plane, whose shapes are defined by their equations.

Example:

The parabola y=x² and the line y=x are examples of Curves whose enclosed area can be calculated.

Definite Integral

Criticality: 3

The integral of a function over a specified interval, representing the net area between the function and the x-axis.

Example:

Definite Integral

Criticality: 3

An integral that computes the net accumulation of a quantity, such as the area under a curve or between curves, over a specific interval defined by upper and lower limits.

Example:

Calculating the Definite Integral of a velocity function from t=0 to t=5 gives the total displacement over that time.

Definite Integral

Criticality: 3

A mathematical operation that calculates the net signed area between a function's graph and the x-axis over a specified interval.

Example:

Using a definite integral from 0 to 5 for a velocity function $v(t)$ gives the total displacement of an object over that time.

Definite Integral

Criticality: 3

An integral with specified upper and lower limits, used to calculate the net area under a curve or between curves over a given interval.

Example:

Using a definite integral like $\int_{1}^{3} x^2 , dx$ to find the exact area under the parabola from $x=1$ to $x=3$ .

Function Order (Above/Below Function)

Criticality: 3

The principle of identifying which function has greater y-values (or x-values for integration with respect to y) over a given interval, ensuring the integrand is non-negative.

Example:

When integrating with respect to x, always place the Function Order of the curve that is graphically above the other first in the integrand to ensure a positive area.

Function x = g(y)

Criticality: 3

A mathematical relationship where the independent variable is $y$ and the dependent variable is $x$, meaning $x$ is expressed in terms of $y$.

Example:

To find the area with respect to the y-axis, you must first rearrange $y = x^2 - 4$ into the form of a function x = g(y), which would be $x = \pm\sqrt{y+4}$ .

Integrand

Criticality: 2

The function that is being integrated within an integral expression.

Example:

In the integral ∫(x² + 3x) dx, the expression (x² + 3x) is the Integrand.

Integration Limits

Criticality: 3

The upper and lower bounds of a definite integral, which define the specific interval over which the integration is performed.

Example:

When finding the area between $x=y^2$ and the y-axis from $y=1$ to $y=3$ , $y=1$ and $y=3$ are the integration limits.

Integration with respect to x

Criticality: 3

The process of finding the area between curves by setting up the integral in terms of the variable x, typically when functions are given as y = f(x).

Example:

To find the area between y=x² and y=x+2, you would use Integration with respect to x, setting up the integral as ∫(x+2 - x²) dx.

Integration with respect to y

Criticality: 3

The process of finding the area between curves by setting up the integral in terms of the variable y, typically when functions are given as x = f(y).

Example:

If you have curves like x=y² and x=y+2, you would use Integration with respect to y, setting up the integral as ∫(y+2 - y²) dy.

Intersection Points

Criticality: 3

Points where two curves meet or cross each other, which are critical for determining the limits of integration when calculating areas between curves.

Example:

To find the area enclosed by the parabola y = x^2 and the line y = x + 2, you first identify their intersection points to define the boundaries of integration.

Intersection Points

Criticality: 3

The specific coordinates where two or more curves meet or cross each other, often serving as the limits of integration when finding the area between them.

Example:

Finding the Intersection Points of y=x² and y=4 is the first step to determine the bounds for calculating the area enclosed by them.

Interval

Criticality: 2

A continuous range of values for a variable, often denoted by [a, b], over which a function is considered or an integral is evaluated.

Example:

When finding the area between two curves from x=0 to x=2, the Interval is [0, 2].

Limits (of integration)

Criticality: 3

The upper and lower bounds of an interval over which a definite integral is calculated, defining the specific region of interest.

Example:

When finding the area under $y=x^2$ from $x=1$ to $x=3$ , 1 and 3 are the limits of integration.

Logarithms

Criticality: 2

The inverse operation to exponentiation, used to find the power to which a base number must be raised to produce a given number.

Example:

When solving for x in 2^x = 8, you use Logarithms to find that x = log₂(8) = 3.

Multiple Integrals

Criticality: 2

Example:

When finding the area of a region bounded by three different functions, you might need to use multiple integrals, each with different limits and integrand functions, to cover the entire region.

Negative Area Integrals

Criticality: 2

A definite integral whose value is negative, indicating that the area lies predominantly to the left of the y-axis (or below the x-axis for $\int f(x) dx$).

Example:

If you integrate $x = y^2 - 9$ from $y=0$ to $y=2$ , the result will be a negative area integral because the curve is to the left of the y-axis in that interval.

Negative Areas

Criticality: 3

A result of a definite integral when the function's graph lies below the x-axis over the interval, indicating a signed area rather than a physical area.

Example:

If a definite integral for a profit function yields a negative area, it signifies a net loss over that period.

Top Function

Criticality: 2

In the context of finding the area between two curves, the function with the higher y-value (or further to the right for integration with respect to y) within a specified interval.

Example:

When finding the area between y = √x and y = x^2 from x=0 to x=1, y = √x is the top function because its values are greater than or equal to y = x^2 in that interval.

Total Areas Between Two Curves

Criticality: 3

A method for calculating the total area enclosed by two intersecting curves, typically involving integrating the difference between the 'top' and 'bottom' functions over relevant intervals.

Example:

Determining the land area shared by two overlapping circular fields would involve calculating the total areas between two curves representing the field boundaries.

Y-Axis Intercepts

Criticality: 2

The points where a graph crosses or touches the y-axis, occurring when the x-coordinate is zero.

Example:

For the curve $x = y^2 - 4$ , the y-axis intercepts are found by setting $x=0$ , yielding $y=\pm 2$ .

x-axis intercepts

Criticality: 2

The points where a function's graph crosses or touches the x-axis, meaning the y-coordinate is zero.

Example:

Finding the x-axis intercepts of a quadratic function helps determine the starting and ending points for calculating the area it encloses with the x-axis.