Definite Integrals and Accumulation

A reliable method

1. Identify the integrand and ordered bounds.
2. Find an antiderivative.
3. Evaluate at the upper bound and subtract the lower-bound value.
4. Interpret sign and units; net accumulation differs from total unsigned area.

Example 1

Evaluate the definite integral \(\int_{0}^{3}(2 x)\,dx\).

1. Find an antiderivative: \(F(x)=x^{2}\).
2. By the Fundamental Theorem of Calculus, evaluate F at the upper endpoint and subtract F at the lower endpoint.
3. \(9-(0)=9\). This is a signed net integral; it equals geometric area only when the relevant sign conditions hold.

Answer: \(9\)

Example 2

Evaluate the definite integral \(\int_{0}^{2}(x^{2})\,dx\).

1. Find an antiderivative: \(F(x)=\frac{x^{3}}{3}\).
2. By the Fundamental Theorem of Calculus, evaluate F at the upper endpoint and subtract F at the lower endpoint.
3. \(\frac{8}{3}-(0)=\frac{8}{3}\). This is a signed net integral; it equals geometric area only when the relevant sign conditions hold.

Answer: \(\frac{8}{3}\)

Common mistakes

• A definite integral has no remaining arbitrary constant.
• Reversing bounds reverses the sign.

Practice without the answer first

1. Evaluate the definite integral \(\int_{1}^{3}(3 x^{2} - 2)\,dx\).
2. Evaluate the definite integral \(\int_{-1}^{1}(x)\,dx\).
3. Evaluate the definite integral \(\int_{2}^{5}(4)\,dx\).

These original skill exercises are not official exam questions or a score prediction. Use the coverage notes on your exam hub to choose relevant practice.