Riemann Sum Practice

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 smaller sub-intervals and summing the areas of rectangles formed by the function values at specific points.

Front

What is the difference between Left Riemann Sum and Right Riemann Sum?

Back

In a Left Riemann Sum, the height of each rectangle is determined by the function value at the left endpoint of each sub-interval, while in a Right Riemann Sum, the height is determined by the function value at the right endpoint.

Front

What is the Trapezoidal Rule?

Back

The Trapezoidal Rule is a numerical method for estimating the integral of a function by approximating the area under the curve as a series of trapezoids rather than rectangles.

Front

How do you calculate the area using Left Riemann Sum?

Back

To calculate the area using Left Riemann Sum, multiply the width of each sub-interval by the function value at the left endpoint and sum these products.

Front

What is the formula for the Trapezoidal Rule?

Back

The formula for the Trapezoidal Rule is: T = (b-a)/(2n) * (f(a) + 2 * Σf(x_i) + f(b)), where n is the number of sub-intervals.

Front

What does 'n' represent in Riemann Sums?

Back

In Riemann Sums, 'n' represents the number of sub-intervals into which the total interval is divided.

Front

How do you determine the width of each sub-interval in Riemann Sums?

Back

The width of each sub-interval is determined by the formula: Δx = (b - a) / n, where [a, b] is the interval and n is the number of sub-intervals.

Front

What is the significance of the function values in Riemann Sums?

Back

The function values at specific points determine the height of the rectangles used to approximate the area under the curve.

Front

How do you approximate the integral using the Trapezoidal Rule?

Back

To approximate the integral using the Trapezoidal Rule, calculate the area of each trapezoid formed by the function values at the endpoints of the sub-intervals and sum these areas.

Front

What is the purpose of using numerical methods like Riemann Sums and the Trapezoidal Rule?

Back

Numerical methods like Riemann Sums and the Trapezoidal Rule are used to estimate the area under curves when an exact integral cannot be easily computed.