Fundamental Theorem of Calculus, Part 2

Definition

The Fundamental Theorem of Calculus, Part 2 states that if $F$ is an antiderivative of a continuous function $f$ on an interval $[a, b]$, then the integral of $f$ from $a$ to $b$ is given by $F(b) - F(a)$. It connects differentiation with integration, showing that these two operations are essentially inverses of each other.

5 Must Know Facts For Your Next Test

1. The theorem requires the function $f$ to be continuous on the closed interval $[a, b]$.
2. If $F'$ exists and equals $f$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.
3. This theorem implies that the process of finding the area under a curve can be simplified using antiderivatives.
4. It is used to evaluate definite integrals without having to compute Riemann sums directly.
5. The Fundamental Theorem of Calculus consists of two parts: Part 1 deals with differentiating an integral, while Part 2 deals with evaluating a definite integral.

Review Questions

Related terms

Continuous Function:

A function is continuous on an interval if it is uninterrupted or unbroken at every point within that interval.

Antiderivative:

A function whose derivative is the original function. Also known as an indefinite integral.

Definite Integral:

$\int_a^b f(x) \, dx$, representing the signed area under the curve $f(x)$ from $x=a$ to $x=b$.