Moment of a Region About the X-Axis Calculator

Enter the coefficients of f(x) in descending power order and provide the lower and upper bounds. The calculator squares the polynomial, integrates the result exactly term by term, and multiplies by one half. It returns the moment and original polynomial degree as structured values suitable for checking calculus exercises or continuing a centroid calculation.

Describe the curve and interval unambiguously

Begin by writing the boundary curve as a polynomial f(x), then enter its coefficients from the highest power down to the constant term. Keep a zero for every missing power: x cubed minus 2x is represented by [1, 0, -2, 0]. This ordered representation avoids the ambiguities that formula text can introduce through exponent symbols, implicit multiplication, variable names, or parentheses. Next, supply the lower and upper bounds of the closed interval, with the upper bound strictly greater than the lower bound. The intended geometric picture is a region between the curve and the x-axis, bounded at both ends by vertical lines. Although squaring makes the integral nonnegative even where f changes sign, a region described as being under one curve normally assumes that the selected boundary is appropriate throughout the interval. Check a sketch or evaluate several points if the polynomial crosses the axis. If crossings divide the picture into separate regions, choose bounds that match the particular region whose moment you need. Coefficients and endpoints must be finite numbers, and the coefficient list may contain at most one hundred entries.

Understand the one-half squared-function formula

For a thin vertical strip at position x, extending from y equals zero to y equals f(x), the moment about the x-axis is found by integrating y across the strip before integrating over x. The inner calculation gives one half times f(x) squared, so the full expression is Mx = one half times the integral from a to b of [f(x)] squared dx. The calculator performs that operation analytically for polynomial input. It first multiplies the coefficient array by itself, which produces the coefficients of the squared polynomial without sampling or numerical quadrature. It then integrates every resulting monomial using the power rule, evaluates the antiderivative at both interval endpoints, subtracts the lower value from the upper value, and divides the total by two. For example, if f(x) equals x, squaring produces x squared and the moment from zero to two is one half times eight thirds, or four thirds. Results are rounded to fifteen significant digits to keep JSON output stable while retaining useful floating-point precision. No network request, random choice, clock, or saved state affects the calculation.

Use the result in region and centroid problems

The returned moment_x_axis is the first moment of area about the x-axis, not the area itself and not a mechanical mass moment of inertia. Its dimensions are length cubed when x and f(x) use the same length unit. This distinction matters when you continue to the vertical coordinate of a centroid: compute the region's ordinary area A as the integral of f(x), then use y-bar = Mx divided by A, provided the geometric assumptions and units are consistent. The calculator intentionally does not infer units, find intersections, parse symbolic expressions, draw the curve, or calculate the area and centroid automatically. Those steps require separate decisions that should remain visible in a worked solution. A negative polynomial value does not make this formula negative because the boundary is squared, but such a value may indicate that the chosen interval does not describe the region you intended. Very large degrees, coefficients, or endpoints can exceed JavaScript's finite numeric range; in that case the tool rejects the request rather than returning infinity. Through the API, each deterministic calculation costs $0.002, making the same operation convenient for repeatable worksheets and automated pipelines.