Finding limit of sum using Riemann Sums

Thread starter hali
Start date Mar 24, 2013
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Limit Riemann Riemann sums Sum Sums

In summary, a Riemann Sum is a method used to approximate the area under a curve by dividing the region into smaller rectangles and finding the sum of their areas. It is useful to find the limit of a sum using Riemann Sums because it allows for accurate determination of the area under a curve, which has practical applications in various fields. There are two types of Riemann Sums - left and right - which use different endpoints to determine the height of the rectangles. The more subintervals used in a Riemann Sum, the more accurate the approximation will be, but there is a balance between accuracy and efficiency. As the width of the subintervals approaches zero, the Riemann Sum approaches the definite integral

Mar 24, 2013

hali

Find the limit limn→∞∑i=1 i/n^2+i^2 by expressing it as a definite integral of an appropriate
function via Riemann sums

...?

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Mar 24, 2013

HallsofIvy

Science Advisor

Homework Helper

Okay, exactly what difficulty are you having? Do you know what a "Riemann sum" is and how it is connected to an integral?

Mar 24, 2013

hali

Exactly, no idea what it is :/

FAQ: Finding limit of sum using Riemann Sums

What is a Riemann Sum?

A Riemann Sum is a method used to approximate the area under a curve by dividing the region into smaller rectangles and finding the sum of their areas.

Why is it useful to find the limit of a sum using Riemann Sums?

Finding the limit of a sum using Riemann Sums allows us to accurately determine the area under a curve, which has many practical applications in fields such as physics, economics, and engineering.

What is the difference between a left Riemann Sum and a right Riemann Sum?

A left Riemann Sum uses the left endpoint of each subinterval to determine the height of the rectangle, while a right Riemann Sum uses the right endpoint. This can result in slightly different approximations of the area under the curve.

How do you choose the number of subintervals for a Riemann Sum?

The more subintervals used, the more accurate the approximation will be. However, using too many subintervals can be computationally intensive. It is important to find a balance between accuracy and efficiency when choosing the number of subintervals.

What happens to the Riemann Sum as the width of the subintervals approaches zero?

As the width of the subintervals gets smaller, the Riemann Sum approaches the actual area under the curve, known as the definite integral. This is why taking the limit of a Riemann Sum allows us to accurately find the area under a curve.