Double Integral Midpoint Rule Calculator

This double integral midpoint rule calculator approximates the integral of f(x,y) over a rectangular region by dividing both coordinate directions into equal subintervals. It evaluates the function at the center of every resulting cell, multiplies the sum of those values by the area of one cell, and returns the complete calculation. Enter an expression, four bounds, and the grid dimensions to obtain the approximation together with the spacing, sample count, midpoint coordinates, function values, and contribution from each subrectangle.

Set up the rectangle and midpoint grid

A composite midpoint approximation begins with a rectangle [a,b] by [c,d]. The calculator computes the widths dx=(b-a)/m and dy=(d-c)/n. Their product is the area of every subrectangle.

Within column i and row j, the sample coordinate is halfway between the corresponding edges, so no corner or boundary value is used. The integrand can contain x, y, ordinary arithmetic, powers, the constants pi and e, and supported functions such as sin, cos, exp, log, sqrt, abs, min, and max. Multiplication must be explicit: write 2*x rather than 2x.

Bounds must increase from minimum to maximum, and each subdivision count must be a positive integer. These requirements make the region, grid, and orientation unambiguous before any numerical work begins. The reported sample list uses one-based column and row indices, making each value easy to match to a hand-drawn grid or classroom table.

Understand how the approximation is formed

For every grid cell, the calculator evaluates f at that cell's center. It adds all midpoint values and multiplies their sum by dx times dy. Equivalently, each row in the returned sample list includes a contribution equal to the function value multiplied by the common cell area, and adding those contributions produces the displayed approximation.

Midpoint sampling often performs better than choosing a corner because the center balances variation across a small cell, but the result is still a numerical approximation. Accuracy depends on the shape of the function and the grid resolution. Smooth functions generally improve as the subdivisions become finer, while steep gradients, oscillation, singular behavior, or discontinuities may require much more care.

The calculator rejects any midpoint where the expression is undefined or produces an infinite value. It does not claim an error bound or compare against an exact antiderivative. To assess convergence, run the same rectangle with progressively finer grids and compare the approximations.

Agreement across refinements is useful evidence, although it is not by itself a formal proof of accuracy.

Read, verify, and reuse the result

The result separates the inputs, grid geometry, accumulated value sum, final approximation, and individual samples. Start by checking that the returned rectangle and subdivision counts match the intended problem. Next verify dx, dy, and cell area; a mistaken bound or swapped count becomes apparent there.

The sample count must equal the product of the two subdivision counts. Each sample reports its midpoint coordinates, evaluated function value, and weighted contribution, which provides an audit trail for homework, worksheets, numerical-method demonstrations, and software tests. The precision option changes only how returned decimals are presented; evaluations and accumulation retain the available floating-point precision internally.

A coarse grid is valuable for checking a hand calculation because every term remains visible. A finer grid is better for studying convergence, but it also creates a longer sample list and more computation. No network service, random source, stored state, or clock participates in the calculation, so results are reproducible within standard JavaScript floating-point arithmetic.