Glossary
Accumulations of change
The total amount of change in a quantity over an interval, often represented by the area under a rate-of-change curve.
Example:
Calculating the total distance traveled by a car given its velocity function over time involves finding the accumulations of change in position.
Concave down
A characteristic of a function's graph where its slope is decreasing, causing a trapezoidal sum to underestimate and a midpoint sum to overestimate the area.
Example:
For a function like f(x) = -x^2, which is always concave down, a Trapezoidal Sum will give an underestimate of the area.
Concave up
A characteristic of a function's graph where its slope is increasing, causing a trapezoidal sum to overestimate and a midpoint sum to underestimate the area.
Example:
For a function like f(x) = x^2, which is always concave up, a Trapezoidal Sum will give an overestimate of the area.
Left Riemann Sum
A Riemann sum where the height of each rectangle is determined by the function's value at the left endpoint of its subinterval.
Example:
When approximating the area under f(x) = e^x from x=0 to x=2 with two rectangles, a Left Riemann Sum would use f(0) and f(1) for the heights.
Midpoint Riemann Sum
A Riemann sum where the height of each rectangle is determined by the function's value at the midpoint of its subinterval.
Example:
To approximate the area under f(x) = sin(x) from x=0 to x=pi with two rectangles, a Midpoint Riemann Sum would use f(pi/4) and f(3pi/4) for the heights.
Overestimating
Occurs when a Riemann sum approximation yields a value greater than the true area under the curve.
Example:
A Right Riemann Sum for an increasing function will always overestimate the true area because the rectangles extend above the curve.
Riemann sums
A method for approximating the definite integral of a function by dividing the area under its curve into a series of simple geometric shapes, typically rectangles or trapezoids.
Example:
To estimate the area under y = x^2 from x=0 to x=4, you could use a Riemann sum with four rectangles.
Right Riemann Sum
A Riemann sum where the height of each rectangle is determined by the function's value at the right endpoint of its subinterval.
Example:
For the same f(x) = e^x from x=0 to x=2 with two rectangles, a Right Riemann Sum would use f(1) and f(2) for the heights.
Subdivisions
The smaller intervals created when dividing the total interval over which a Riemann sum is calculated, forming the bases of the approximating shapes.
Example:
If you approximate the area from x=0 to x=10 using 5 rectangles, each rectangle will have a width of 2, representing the subdivisions of the interval.
Trapezoidal Sum
A method for approximating the area under a curve by dividing it into trapezoids, where the parallel sides are the function values at the endpoints of each subinterval.
Example:
Calculating the area under a velocity-time graph using a Trapezoidal Sum often provides a more accurate estimate of total displacement than rectangular sums.
Underestimating
Occurs when a Riemann sum approximation yields a value less than the true area under the curve.
Example:
A Left Riemann Sum for an increasing function will always underestimate the true area because the rectangles fall below the curve.