Area Under Curve Calculator

Calculate definite integrals with step-by-step breakdown. Symbolic & numeric methods.

Fundamental Theorem of Calculus: ∫ab f(x) dx = F(b) - F(a), where F'(x) = f(x).

Use ^ for exponent, * for multiplication, sin(), cos(), exp(), log(), sqrt()
x²
x³ - 3x
sin(x)
cos(x)
eˣ
ln(x)
√x
1/x
4x - x²

? Understanding Area Under a Curve

The area between the graph of a function y = f(x) and the x‑axis, from x = a to x = b, is given by the definite integral. This concept is fundamental in calculus and appears in countless applications.

? Geometric Interpretation

If f(x) ≥ 0 on [a,b], the integral equals the geometric area. If f(x) becomes negative, the integral gives signed area (positive above axis, negative below). To obtain total (absolute) area, integrate |f(x)|.

? Riemann Sum – The Building Block

The definite integral is defined as the limit of Riemann sums:

∫ab f(x) dx = limn→∞ Σi=1n f(xi*) Δx, where Δx = (b−a)/n.

Common choices for the sample point xi* are left endpoint, right endpoint, or midpoint. The animation below (conceptual) shows how rectangles approximate the area.

Left sum : overestimates if decreasing
Right sum : underestimates if decreasing
Midpoint rule : often more accurate
? Properties of the Definite Integral
  • Linearity: ∫ (c·f + g) = c∫ f + ∫ g
  • Interval additivity: ∫ab + ∫bc = ∫ac
  • Comparison: if f(x) ≤ g(x), then ∫ f ≤ ∫ g
  • Zero width: ∫aa f = 0
? The Fundamental Theorem of Calculus (FTC)

Part 1: If F'(x) = f(x), then ∫ab f(x) dx = F(b) - F(a).
Part 2: d/dx [∫ax f(t) dt] = f(x).

The FTC links differentiation and integration, allowing us to compute integrals exactly when an antiderivative exists.

? Numerical Integration (When Antiderivatives Are Hard)

Many functions (e.g., ex², sin(x)/x) lack elementary antiderivatives. Numerical methods approximate the integral:

  • Trapezoidal rule: approximates area using trapezoids.
  • Simpson's rule: uses parabolic arcs; more accurate for smooth functions.
  • Adaptive methods: refine steps in regions where function varies rapidly.
? Real‑World Applications
  • Physics: displacement = ∫ v dt, work = ∫ F dx
  • Probability: area under density curve = probability
  • Economics: consumer surplus, total cost
  • Engineering: total charge Q = ∫ I dt
  • Biology: cardiac output = ∫ concentration
  • Statistics: mean value = (1/(b−a))∫ f

? How this calculator works: It first attempts symbolic integration via nerdamer. If successful, the exact antiderivative and value are shown. If not, it falls back to adaptive Simpson numeric integration (or trapezoidal). The graph always displays the filled area and the limits.

Frequently Asked Questions

The calculator returns the signed area. For total area, integrate |f(x)| (you can enter abs(x) but nerdamer may not handle it; use numeric method or split intervals).

Adaptive Simpson typically achieves 1e-8 relative error for smooth functions. The trapezoidal rule (with 1000 subintervals) is less accurate but still reliable for most purposes.

Many integrands (like e^(x^2), sin(x)/x, or functions with no closed‑form antiderivative) are not expressible in elementary functions. The calculator then automatically switches to numerical methods.