RAM, Geometric Integrals, Average Value

Front

What is a Riemann Sum?

Back

A Riemann Sum is a method for approximating the total area under a curve by dividing the region into rectangles and summing their areas.

Front

What is the difference between Left Riemann Sum (LRAM) and Right Riemann Sum (RRAM)?

Back

LRAM uses the left endpoints of sub-intervals to determine the height of rectangles, while RRAM uses the right endpoints.

Front

What is the Midpoint Riemann Sum (MRAM)?

Back

MRAM uses the midpoints of sub-intervals to determine the height of rectangles, providing a more accurate approximation than LRAM or RRAM.

Front

How do you calculate the area using LRAM?

Back

Area = Σ (height of rectangle) × (width of rectangle) using left endpoints.

Front

For a strictly decreasing function, what type of estimate does RRAM provide?

Back

RRAM provides an overestimate for a strictly decreasing function.

Front

For a strictly increasing function, what type of estimate does LRAM provide?

Back

LRAM provides an underestimate for a strictly increasing function.

Front

What is the formula for calculating the average value of a function over an interval [a, b]?

Back

Average Value = (1/(b-a)) * ∫[a to b] f(x) dx.

Front

What is the significance of the average value of a function?

Back

The average value represents the 'typical' value of the function over the specified interval.

Front

How does the number of sub-intervals affect the accuracy of a Riemann Sum?

Back

Increasing the number of sub-intervals generally increases the accuracy of the Riemann Sum approximation.

Front

What is the relationship between Riemann Sums and definite integrals?

Back

Riemann Sums are used to approximate definite integrals, which represent the exact area under a curve.