Riemann Sum Notation - Pugh

Front

What is the definition of a Riemann Sum?

Back

A Riemann Sum is a method for approximating the total area under a curve by dividing it into rectangles, calculating the area of each rectangle, and summing these areas.

Front

What does the notation \( \lim_{n\rightarrow\infty} \sum_{k=1}^n f(x_k^*) \Delta x \) represent?

Back

This notation represents the definite integral of a function \( f \) from a to b, where \( x_k^* \) is a sample point in each subinterval and \( \Delta x \) is the width of each subinterval.

Front

What is the lower limit in the context of Riemann Sums?

Back

The lower limit is the starting point of the interval over which the function is being integrated.

Front

What does \( a + \frac{b-a}{n}k \) represent in summation notation?

Back

It represents the x-value at the k-th subinterval in the partition of the interval [a, b].

Front

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

Back

The width of each rectangle is determined by \( \Delta x = \frac{b-a}{n} \), where \( n \) is the number of rectangles.

Front

What is the significance of the upper limit in Riemann Sums?

Back

The upper limit is the endpoint of the interval over which the function is being integrated.

Front

What is the integral notation for the summation \( \lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\frac{3}{n}\right)\left[\left(\frac{3k}{n}\right)^3\right] \)?

Back

The integral notation is \( \int_0^3 x^3 dx \).

Front

What is the integral notation for the summation \( \lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\frac{5}{n}\right)\left[4\left(\frac{5k}{n}\right)\right] \)?

Back

The integral notation is \( \int_0^5 4x dx \).

Front

What does \( \Delta x \) represent in the context of Riemann Sums?

Back

\( \Delta x \) represents the width of each rectangle used in the approximation of the area under the curve.

Front

What is the integral notation for the summation \( \lim_{n\rightarrow\infty}\sum_{k=1}^n\left(\frac{\pi}{n}\right)\left[\cos\left(\pi+\frac{\pi k}{n}\right)\right] \)?

Back

The integral notation is \( \int_{\pi}^{2\pi} \cos x dx \).