Centroid X-Coordinate of a Region Calculator

Enter the polynomial as coefficients in descending power order, together with the lower and upper bounds. The calculator evaluates the area integral, computes the first moment about the y-axis, and divides that moment by the area. It returns all three values so you can inspect the work, reuse intermediate results, and confirm the centroid rather than accepting an unexplained coordinate.

Represent the curve and interval correctly

Start by expressing the upper boundary as a polynomial f(x). Supply its coefficients from the highest power down to the constant term, keeping a zero wherever a power is missing. For example, x squared plus 3 is entered as [1, 0, 3], because the coefficient of x is zero. Then provide a lower bound and an upper bound with the upper value strictly greater than the lower value. The intended geometric region lies between this curve and the x-axis across that interval, so the curve should remain above the axis. This calculator does not parse formulas or discover intersections automatically; explicit coefficients and endpoints make the input deterministic and prevent ambiguous interpretations of notation. Before calculating, sketching the curve or checking important values can help confirm that your chosen interval describes one continuous region above the axis. If the curve crosses the axis, split the problem at its roots and decide which bounded region you actually intend to measure instead of treating signed cancellation as ordinary geometric area.

Understand the area and moment calculation

The horizontal centroid coordinate is defined by x-bar equals M sub y divided by A. Here A is the area integral from the lower bound to the upper bound of f(x), and M sub y is the first moment about the y-axis, found by integrating x times f(x) over the same bounds. Multiplication by x gives material farther from the y-axis more influence, which is exactly what a balance coordinate must capture. The solver integrates every polynomial term analytically: a term c times x to the n becomes c divided by n plus 1 times x to the n plus 1 for area, while the moment calculation first raises the term's power by one. It evaluates both antiderivatives at the endpoints and subtracts the lower value from the upper value. Finally, it divides the accumulated moment by the accumulated area. Returning A and M sub y alongside x-bar makes it easy to compare the output with handwritten work and locate an incorrect coefficient or bound.

Interpret and verify the returned coordinate

For a valid region with nonnegative height, the returned x-coordinate should fall between the interval endpoints. Its position reflects how the region's area is distributed horizontally, not where the curve reaches its maximum and not the average of the two bounds. A symmetric region produces the midpoint, while extra area toward one side pulls the centroid in that direction. Use the returned area and first moment as verification data: dividing first_moment_y_axis by area should reproduce x_coordinate. Units also provide a useful check. If x is measured in meters and f(x) in meters, area is measured in square meters, the first moment is measured in cubic meters, and their quotient is measured in meters. The capability rejects reversed or equal bounds, non-finite inputs, numerical overflow, and a nonpositive signed area. A positive result alone does not prove the curve never crossed the axis, however. When geometric validity matters, confirm that f(x) stays at or above zero throughout the selected interval before interpreting the ratio as the centroid of one region under the curve.