Grades Pre-Calculus & Calculus IntegralsDefinite Integrals & the Fundamental Theorem

Definite Integrals & the Fundamental Theorem

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Definite Integrals & the Fundamental Theorem

Compute exact areas using the Fundamental Theorem of Calculus.

The definite integral ∫_a^b f(x) dx represents the net signed area between the curve y = f(x) and the x-axis from x = a to x = b. The Fundamental Theorem of Calculus (FTC) links differentiation and integration — two seemingly unrelated operations.

\int_a^b f(x)\,dx = F(b) - F(a) \quad \text{where } F'(x) = f(x)

\text{FTC Part 1: } \frac{d}{dx}\int_a^x f(t)\,dt = f(x)

A Riemann sum approximates the definite integral by dividing [a, b] into n subintervals of width Δx = (b−a)/n and summing the areas of rectangles. As n→∞, the Riemann sum converges to the exact integral.

\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\,\Delta x

📐Computing a Definite Integral

∫₁³ (x² + 1) dx. F(x) = x³/3 + x. F(3) = 9 + 3 = 12. F(1) = 1/3 + 1 = 4/3. Integral = 12 − 4/3 = 32/3.

\int_1^3 (x^2+1)\,dx = \left[\frac{x^3}{3}+x\right]_1^3 = 12 - \frac{4}{3} = \frac{32}{3}

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Remember This!

The notation [F(x)]_a^b means F(b) − F(a). Always subtract the value at the lower limit from the value at the upper limit, not the other way around.

✏️ Try It!

Evaluate ∫₀² (3x² − 2x) dx.

Practice with CalcVerse

∫Integral Evaluator

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