Glossary
Definite Integral
The exact area under a curve between two specified points, defined as the limit of a Riemann sum as the number of subintervals approaches infinity.
Example:
The definite integral ∫₀⁵ x² dx calculates the precise area under the parabola y = x² from x=0 to x=5.
Delta x (Δx)
The constant width of each subinterval in a Riemann sum, calculated as (b-a)/n.
Example:
For a Riemann sum on [0, 10] with n=5 subintervals, Δx would be (10-0)/5 = 2.
Left Riemann Sum
A type of Riemann sum where the height of each rectangle is determined by the function's value at the left endpoint of its corresponding subinterval.
Example:
Approximating the area under f(x) = x² on [0, 2] using two rectangles and taking the height from the left side of each interval results in a Left Riemann Sum.
Limit (as n approaches infinity)
A mathematical concept used to define the definite integral, where the number of subintervals in a Riemann sum becomes infinitely large, leading to the exact area.
Example:
The limit as n approaches infinity of a Riemann sum lim_{n→∞} Σ f(xᵢ)Δx precisely defines the definite integral.
Lower Bound (a)
The starting x-value of the interval over which the area under a curve is being calculated or approximated.
Example:
In the integral ∫₁³ f(x) dx, the value 1 is the lower bound of integration.
Riemann Sum
An approximation of the area under a curve by dividing the region into a series of rectangles and summing their areas.
Example:
Using four rectangles to estimate the area under f(x) = x² from x=0 to x=4 is an example of a Riemann Sum.
Right Riemann Sum
A type of Riemann sum where the height of each rectangle is determined by the function's value at the right endpoint of its corresponding subinterval.
Example:
Approximating the area under f(x) = x² on [0, 2] using two rectangles and taking the height from the right side of each interval results in a Right Riemann Sum.
Subinterval (n)
One of the equal-width divisions of the total interval over which the area under a curve is being approximated in a Riemann sum.
Example:
If you divide the interval [0, 10] into 5 equal parts for a Riemann sum, then n=5 and each part is a subinterval.
Summation Notation (Σ)
A concise way to represent the sum of a sequence of terms, often used to express Riemann sums algebraically.
Example:
The expression Σ_{i=1}^{5} i² represents the sum of the first five perfect squares, which is a form of summation notation.
Upper Bound (b)
The ending x-value of the interval over which the area under a curve is being calculated or approximated.
Example:
In the integral ∫₁³ f(x) dx, the value 3 is the upper bound of integration.
x_i
The x-coordinate used to determine the height of the *i*-th rectangle in a Riemann sum, typically an endpoint or midpoint of the *i*-th subinterval.
Example:
For a right Riemann sum starting at a=0 with Δx=0.5, the third x_i would be 0 + 0.53 = 1.5.