Riemann Sums

Front

What is a Riemann Sum?

Back

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

Front

What is the formula for a Riemann Sum using left endpoints?

Back

Front

What is the difference between left-endpoint and right-endpoint Riemann Sums?

Back

Left-endpoint sums use the left side of each subinterval to determine the height of the rectangles, while right-endpoint sums use the right side.

Front

How does increasing the number of rectangles affect the accuracy of a Riemann Sum?

Back

Increasing the number of rectangles generally increases the accuracy of the Riemann Sum, as it better approximates the area under the curve.

Front

What is the significance of the interval in a Riemann Sum?

Back

The interval defines the range over which the function is being integrated and affects the width of the rectangles used in the approximation.

Front

What is the relationship between Riemann Sums and definite integrals?

Back

Riemann Sums are used to approximate definite integrals, and as the number of rectangles approaches infinity, the Riemann Sum approaches the exact value of the definite integral.

Front

What is the Midpoint Riemann Sum?

Back

The Midpoint Riemann Sum uses the midpoint of each subinterval to determine the height of the rectangles, providing a different approximation than left or right endpoint sums.

Front

How do you determine the width of each rectangle in a Riemann Sum?

Back

Front

What is the effect of using trapezoids instead of rectangles in approximating an integral?

Back

Using trapezoids can provide a more accurate approximation of the area under the curve compared to rectangles, as it accounts for the slope of the function.

Front

What is the purpose of using sigma notation in Riemann Sums?

Back

Sigma notation compactly represents the sum of the areas of the rectangles, making it easier to express and calculate Riemann Sums.