Definite Integral via FTC - Calculus Step by Step

Evaluate ∫ x² dx from 1 to 3 using the Fundamental Theorem of Calculus: F(3) - F(1) = 9 - 1/3 = 26/3.

definite-integral-ftc The Fundamental Theorem of Calculus states that ∫_a^b f(x) dx = F(b) - F(a), where F is any antiderivative of f.

Example

Use F of the upper bound minus F of the lower bound. What guarantees this evaluate-the-antiderivative shortcut equals the accumulated area is the Fundamental Theorem of Calculus, which this book uses but does not prove.

highlighted = computed this step

Step 1 — Set up

Set up the definite integral from 1 to 3.

\int_{ 1 }^{ 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 the upper bound 3 to get 9.

F( 3 )= \hl{9}

Step 4 — Evaluate lower bound

Evaluate F at the lower bound 1 to get 1 over 3.

F( 1 )= \hlmath{\frac{1}{3}}

Step 5 — Subtract

Subtract 9 minus 1 over 3 to get 26 over 3.

9 - \frac{1}{3} = \hlmath{\frac{26}{3}}