4.4.1: Resources and Key Concepts — Read

From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

Resources

Integral Function (Area Function): A function \(F(x)\) defined as a definite integral with a variable upper limit, such as \(F(x) = \displaystyle \int_a^x f(t) \, dt\). It represents the definite integral of a function \(f(t)\) over an interval determined by the argument of \(F\), \(x\).
Evaluation Theorem: Another name for the Fundamental Theorem of Calculus, Part 2.
Notation \(F(x)|_a^b\): Represents the expression \(F(b) - F(a)\).

Theorem: The Fundamental Theorem of Calculus, Part 1 (FTC1): If \(f(x)\) is continuous over an interval \([a,b]\), and the function \(F(x)\) is defined by \(F(x) = \displaystyle \int_a^x f(t) \, dt\), then \(F^{\prime}(x) = f(x)\) over \([a,b]\).
Corollary: General Form of the FTC1: If \(f(x)\), \(U(x)\), and \(L(x)\) are continuous over an interval \([a,b]\), and the function \(F(x)\) is defined by \(F(x) = \displaystyle \int_{L(x)}^{U(x)} f(t) \, dt\), then \(F^{\prime}(x) = f(U(x))U^{\prime}(x) - f(L(x))L^{\prime}(x)\).
Theorem: The Fundamental Theorem of Calculus, Part 2 (The Evaluation Theorem): If \(f(x)\) is continuous over the interval \([a,b]\) and \(F(x)\) is any antiderivative of \(f(x)\), then \(\displaystyle \int_a^b f(x) \, dx = F(b) - F(a)\).

Forgetting the Chain Rule with FTC1: When the upper limit of integration is a function \(u(x)\) (not just \(x\)), forgetting to multiply by \(u^{\prime}(x)\) when finding the derivative of the integral. For example, \(\frac{d}{dx} \displaystyle \int_a^{u(x)} f(t) \, dt = f(u(x))u^{\prime}(x)\), not just \(f(u(x))\).
Incorrectly Handling Two Variable Limits of Integration (FTC1): When both limits of integration are functions of \(x\), \(F(x) = \displaystyle \int_{L(x)}^{U(x)} f(t) \, dt\), applying the corollary correctly: \(F^{\prime}(x) = f(U(x))U^{\prime}(x) - f(L(x))L^{\prime}(x)\).
Errors in Finding Antiderivatives (FTC2): Making mistakes when determining an antiderivative \(F(x)\) for the integrand \(f(x)\).
Incorrectly Evaluating \(F(b)-F(a)\) (FTC2):
- Calculating \(F(a)-F(b)\) instead of \(F(b)-F(a)\).
- Algebraic errors during the substitution and subtraction.
Forgetting the Constant of Integration \(C\) in Antiderivatives (FTC2 Context): While any antiderivative works for FTC2, and choosing \(C=0\) is simplest, understanding that the \(C\) would cancel out is important. The text notes that the "\(+C\)" term is not included in the antiderivative for FTC2 because any antiderivative works, and the constant term would be canceled.
Confusing the Two Parts of FTC: Misunderstanding which part applies to differentiating an integral (Part 1) versus evaluating a definite integral using an antiderivative (Part 2).

4.4.1: Resources and Key Concepts is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.