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Section 3.3 The Fundamental Theorem of Calculus
Objectives
-
Relate accumulated area to antiderivatives through the Fundamental Theorem of Calculus.
-
Evaluate definite integrals by subtracting antiderivative values at endpoints.
-
Interpret indefinite integrals as families of antiderivatives.
-
Come back to the area of a circle?
-
FTC part I (Evaluation Theorem)
If
\(f^{(-1)}\) is an antiderivative of
\(f(x)\text{,}\) then:
Aside
\begin{equation*}
\int_{a}^{b} f(x)\dd{x}=f^{(-1)}(b)-f^{(-1)}(a)\text{.}
\end{equation*}
-
FTC part II (Antiderivative Construction Theorem):
If \(f\) is continuous on \([a,b]\text{,}\) then the function
\begin{equation*}
\int_{c}^{x}f(t)\dd{t}
\end{equation*}
is an antiderivative of \(f\text{,}\) that is,
\begin{equation*}
\dv{x}\int_{c}^{x} f(t)\dd{t}=f(x)\text{.}
\end{equation*}
-
If \(f^{(-1)}\) is an antiderivative of \(f\text{,}\) we define
\begin{equation*}
\int f(x)\dd{x}=f^{(-1)}(x)+C
\end{equation*}
where \(C\) is an arbitrary constant. Hence the indefinite integral is the entire family of antiderivatives, all offset from each other by a constant.
-
Practice with antiderivatives, only using the Anti-Power Rule:
\begin{equation*}
\int x^{n}\dd{x}=\frac{1}{n+1}x^{n+1}+C
\end{equation*}
(as long as \(n\neq -1\))
