riemann_prisms3d: Riemann rectangular prisms over a planar region

Description

Approximates the double integral of a scalar function over a planar region using an N-by-M rectangular partition and rectangular prisms with constant height on each cell. The region is defined by an x-interval and two functions giving the lower and upper y-limits. Each valid cell produces a prism whose height corresponds to a chosen estimate of the function on that cell: lower value, upper value, or mean value.

When plot = TRUE, a 3D visualization of the prisms is produced using plotly. Optionally, the actual surface z = F(x, y) can also be drawn over the rectangular bounding box that contains the region.

Usage

riemann_prisms3d(
  f,
  f1,
  f2,
  a,
  b,
  N,
  M,
  plot = TRUE,
  estimate = c("lower", "upper", "mean", "all"),
  sample_n = 6,
  show_surface = FALSE,
  surface_colorscale = "Viridis",
  surface_opacity = 0.35,
  show_surface_grid = TRUE,
  surface_grid_color = "rgba(60,80,200,0.25)",
  surface_grid_width = 1,
  color_by = c("mean", "lower", "upper"),
  top_colorscale = "YlOrBr",
  top_opacity = 0.85,
  side_color = "rgba(60,60,60,0.25)",
  side_opacity = 0.35,
  frame_color = "rgba(0,0,0,0.55)",
  frame_width = 1.5,
  scene = list(aspectmode = "data", xaxis = list(title = "x"), yaxis = list(title = "y"),
    zaxis = list(title = "z")),
  bg = list(paper = "white", plot = "white")
)

Value

A list containing:

sum_lower: Lower Riemann sum for the chosen partition.
sum_upper: Upper Riemann sum.
sum_mean: Mean-value Riemann sum.
cells: A tibble describing all valid cells.
fig: A plotly object if plot = TRUE, otherwise NULL.
estimate: When estimate != "all", the selected Riemann sum (lower/upper/mean) is repeated here for convenience.

Arguments

f: A function f(x, y) returning a numeric scalar.
f1, f2: Functions returning the lower and upper y-boundaries for each x. The valid y-range at each x is the interval between the minimum and maximum of these two functions.
a, b: Numeric endpoints of the x-interval. Must satisfy a < b.
N, M: Integer numbers of subdivisions in x and y for the rectangular partition.
plot: Logical. If TRUE, the 3D visualization is generated.
estimate: Character. One of "lower", "upper", "mean", or "all", indicating which estimate to highlight.
sample_n: Integer. Number of evaluation points per direction inside each cell when computing lower, upper, and mean values.
show_surface: Logical. If TRUE, draws the surface z = F(x, y) over the entire rectangular bounding box.
surface_colorscale: Colorscale used for the surface.
surface_opacity: Opacity for the surface.
show_surface_grid: Logical. If TRUE, draws a grid on the surface.
surface_grid_color: Color of the grid lines.
surface_grid_width: Width of the grid lines.
color_by: Character. Determines the value used to color the top of each prism: "mean", "lower", or "upper".
top_colorscale: Colorscale for the prism tops.
top_opacity: Opacity of the prism tops.
side_color: Color for the vertical faces of the prisms.
side_opacity: Opacity of the prism sides.
frame_color: Color for the prism frame lines.
frame_width: Width of the frame lines.
scene: A list with plotly 3D scene settings.
bg: A list with background colors for the figure.

Examples

Run this code

f  <- function(x, y) x * y
f1 <- function(x) 0
f2 <- function(x) 1 - x
riemann_prisms3d(f, f1, f2, 0, 1, N = 10, M = 10, plot = FALSE)

Run the code above in your browser using DataLab