Lab 14: Riemann Sums¶

In this lab we will study Riemann sums in higher dimensions.

The following is an example of a one dimensional (x) Riemann sum, which you are probably familiar with from Calculus 2. One dimensional Riemann sums are used to approximate two dimensional objects (x, y).

This can be expanded to two dimensions (x, y), to approximate three dimensional objects (x, y, z).

Riemann sums of three dimensions ore more are difficult to visualize. Check out this Wikipedia article about Tesseracts if you want to learn more about four dimensional objects.

Task 1¶

Write a function, riemann_sum_1D(f, x_min, x_max, N, method), that takes as input a function of one variable (x), f, a minimum value of x, x_min, a maximum value of x, x_max, a number of sub-intervals to compute, N, and a Riemann method to use, method ("left", "right", "mid"), and computes the Riemann approximation of f on the input space [x_min, x_max].

Remember the formula for a one dimensional Riemann sum:

\[\sum_{i=1}^N f(x_i^*) \Delta x,\]

where Δx = (xMax - xMin)/N.

Raise a ValueError if the method is not "left", "right", or "mid" (use the error message f"Error: method ({method}) must be \"left\", \"right\", or \"mid\"").

Task 2¶

Write a function, riemann_sum_2D(f, x_min, x_max, y_min, y_max, N, method), that takes as input a function of two variables (x and y), f, a minimum value of x, x_min, a maximum value of x, x_max, a minimum value of y, y_min, a maximum value of y, y_max, a number of sub-intervals to compute, N, and Riemann method to use, method ("left", "right", mid).

Remember the formula for a two dimensional Riemann sum:

\[\sum_{j=1}^N \sum_{i=1}^N f(x_i^*, y_j^*) \Delta x \Delta y,\]

where Δx = (xMax - xMin)/N, Δy = (yMax - yMin)/N

Raise a ValueError if the method is not "left", "right", or "mid" (use the error message f"Error: method ({method}) must be \"left\", \"right\", or \"mid\"").

Task 3¶

Using your function from Task 2, use the midpoint method to calculate the Riemann sums for N=10 and N=20 for the following functions and domains:

\(f(x,y) = x \cdot sin(x \cdot y)\) on \([0, pi] \times [0, pi]\)
\(g(x,y) = y^2 \cdot exp(-x-y)\) on \([0, 1] \times [0, 1]\)
\(h(x,y) = x^3 \cdot y^2 + x \cdot y\) on \([0, 1] \times [1, 2]\)

For each option, print the following information:

f: "function"
xs: "x domain"
ys: "y domain"
N: "number of rectangular prisms"
Riemann sum: "sum"

Replace everything in "" with the actual value. Use f"{number:.5f}"` to show only the first five decimal places of the sum.

Task 4¶

Consider the integral of \(f(x, y) = x \sin(x + y)\) on the rectangle \([0, \pi/6] \times [0, \pi/3]\). First calculate the exact value of this integral by hand. Then make a plot that shows the error of the midpoint Riemann integral approximation as \(N\) ranges from \(1\) to \(100\). Remember to give your graph a title and label your axes.

Task 5¶

Write a function, riemann_sum_3D(f, x_min, x_max, y_min, y_max, z_min, z_max, N, method), that takes as input a function of three variables (x, y, and z), f, a minimum value of x, x_min, a maximum value of x, x_max, a minimum value of y, y_min, a maximum value of y, y_max, a minimum value of z, z_min, a maximum value of z, z_max, a number of sub-intervals to compute, N, and Riemann method to use, method ("left", "right", "mid").

Remember the formula for a three dimensional Riemann sum:

\[\sum_{k=1}^N \sum_{j=1}^N \sum_{i=1}^N f(x_i^*, y_j^*, z_k^*) \Delta x \Delta y \Delta z,\]

where Δx = (xMax - xMin)/N, Δy = (yMax - yMin)/N, Δz = (zMax - zMin)/N.

Raise a ValueError if the method is not "left", "right", or "mid" (use the error message f"Error: method ({method}) must be \"left\", \"right\", or \"mid\"").

Task 6¶

Using your function from Task 5, use the midpoint method to calculate the Riemann sums for N = 10 and N = 20 for the following function and domain:

\[f(x,y,z) = xy+z^2 \text{ on the rectangle } [0,2] \times [0,1] \times [0,3].\]

For each option, print the following information:

f: "function"
xs:     "x domain"
ys: "y domain"
N: "number of rectangular prisms"
Riemann sum: "sum"

Replace everything in "" with the actual value. Use f"{number:.5f}"` to show only the first five decimal places of the sum.

Task 7¶

Write a function, riemann_sum(f, x_min, x_max, N, method), that takes as input a function of an n-dimensional variable (x), f, a vector of minimum values of the components of x, x_min, a vector of maximum values of the components of x, x_max, a number of sub-intervals to compute, N, and Riemann method to use, method ("left", "right", "mid").

Raise a ValueError if the method is not "left", "right", or "mid" (use the error message f"Error: method ({method}) must be \"left\", \"right\", or \"mid\"").