Riemann
Basics
Basics
When we draw a Riemann sum, we represent the area using this common four-sided geometric shape.
A Rectangle
This "S" shaped symbol is used to indicate that you are taking an integral.
Integral Sign
If you have 4 rectangles spanning an interval from x=0 to x=8, what is the width (Δx) of each rectangle?
2
True or False: As you add more rectangles to a Riemann sum, the estimate usually gets more accurate.
True
If the y-axis is "Miles per Hour" and the x-axis is "Hours," the area under the curve represents this.
Total miles (or distance)
If the rectangles in a Riemann sum stick out above the curve, the sum is called an ______-estimate.
Over-Estimate
In the expression ∫f(x)dx, this two-letter symbol represents a "tiny change in or with respect to x."
dx
Find the area of a single Riemann rectangle if its width is 3 and its height (function value) is 5.
15
True or False: The area "under" a curve can never be a negative number in calculus.
False (Area below the x-axis is cons negative in integrals)
If you are integrating a "Rate of Flow" (like gallons per minute), the area represents the total ______ of liquid.
Volume
This type of Riemann sum uses the value of the function at the start of each subinterval to determine rectangle height.
Left Riemann Sum
In a Riemann sum, this Greek letter delta followed by x represents this specific part of each approximating rectangle.
The Width
Calculate 0∫52dx (The area of a rectangle with height 2 and width 5).
10
True or False: A Midpoint Riemann sum uses the middle of the interval to find the height.
True
In physics, the area under a Force vs. Distance graph represents this "W" word.
Work
This approximation method, technically not a Riemann sum, uses the average of the left and right endpoints to form its shape.
The Trapezoidal Rule
When we write 1∫5f(x)dx, the numbers 1 and 5 are called the "lower" and "upper" ______.
Limits (or bound)
If the area under a curve from 0 to 10 is 25, and the area from 0 to 4 is 10, what is the area from 4 to 10?
15
True or False: a∫af(x)dx always equals zero.
True (There is no width)
This word describes the process of finding the area under a curve.
Integration
What is the Trapezoidal Sum formula used to calculate area under the curve?
A=1/2(base)(h1+h2)
The Greek letter Σ (Sigma) in a Riemann sum tells you to perform this mathematical operation on all the rectangles.
Addition (or summation)
If f(x)=x, what is the height of the rectangle in a Right Riemann sum for the interval [2,3]?
3
True or False: If a curve is a straight horizontal line, the Riemann sum will give the exact area regardless of how many rectangles you use.
True
If a car's velocity is constant, the area under its velocity-time graph forms this geometric shape.
A Rectangle