Area Under a Curve - Calculus Step by Step

Find the area under f(x) = x² from x = 0 to x = 3 using the Fundamental Theorem of Calculus: area = F(3) - F(0) = 9.

Example

Use the Fundamental Theorem of Calculus to compute an exact area.

highlighted = computed this step

Step 1 — Set up

Set up the area integral from 0 to 3.

A=\int_{ 0 }^{ 3 } x^{2} \,dx

Step 2 — Antiderivative

Find an antiderivative.

F(x)= \hlmath{\frac{x^{3}}{3}}

Step 3 — Evaluate upper bound

Evaluate F at 3 to get 9.

F( 3 )= \hl{9}

Step 4 — Evaluate lower bound

Evaluate F at 0 to get 0.

F( 0 )= \hl{0}

Step 5 — Area

Subtract 9 minus 0 to get area 9.

9 - 0 = \hl{9}

area-under-curve The area under f(x) from x = a to x = b (where f ≥ 0) equals the definite integral ∫_a^b f(x) dx = F(b) - F(a), where F is any antiderivative of f.