Integral Applications Calculator

None Numerical answer only.

Summary Pill list of all steps applied.

Detailed Full accordion: formula → integrand → numerical evaluation.

A = ∫f dx

Signed definite integral of f(x)

Area Under Curve

A = ∫|f−g|

Handles sign changes automatically

Area Between

L = ∫√(1+(f')²)

Exact curve length along [a,b]

Arc Length

V = π∫R² dx

Rotate about x-axis, single function

Volume (Disk)

V = π∫(R²−r²)

Rotate with inner hole (two radii)

Volume (Washer)

V = 2π∫xf dx

Rotate about y-axis

Volume (Shell)

S = 2π∫f√…

Area of revolution surface

Surface Area

W = ∫F dx

Variable force over displacement

Work

f̄ = ∫f/(b−a)

Mean of f over interval

Average Value

(x̄,ȳ)

Center of mass of 2D region

Centroid

Area \(A=\int_a^b f(x)\,dx\)

Area Between \(A=\int_a^b|f-g|\,dx\)

Arc Length \(L=\int_a^b\sqrt{1+(f'(x))^2}\,dx\)

Disk Volume \(V=\pi\int_a^b[f(x)]^2\,dx\)

Washer Volume \(V=\pi\int_a^b(R^2-r^2)\,dx\)

Shell Volume \(V=2\pi\int_a^b x\,f(x)\,dx\)

Surface Area \(S=2\pi\int_a^b f\sqrt{1+(f')^2}\,dx\)

Work \(W=\int_a^b F(x)\,dx\)

Average Value \(\bar{f}=\frac{1}{b-a}\int_a^b f\,dx\)

Centroid x̄ \(\bar{x}=\frac{\int x(f-g)\,dx}{\int(f-g)\,dx}\)

Centroid ȳ \(\bar{y}=\frac{\int\frac{1}{2}(f^2-g^2)\,dx}{\int(f-g)\,dx}\)

All results are numerical (Gaussian quadrature)
Volume Disk/Shell: enter f(x) only
Volume Washer: enter outer R(x) and inner r(x)
Area Between: enter f(x) top and g(x) bottom (handles crossings)
Centroid: enter f(x) upper boundary; g(x) lower (optional, defaults to 0)
Work: F(x) is force as function of position
Use 3.14159 or pi for π in bounds

∫Integral Applications