Fundamental theorem of calculus and definite integrals

Here are the key points to focus on when studying the Fundamental Theorem of Calculus (FTC) and definite integrals:

1. Definition of Derivative and Integral:

   • Understand the relationship between differentiation and integration.
   • The derivative measures the rate of change, while the integral represents the accumulation of quantities.

2. Fundamental Theorem of Calculus:

   • Part 1: If $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$ on that interval, then:
     $$\int_a^b f(x) \,dx = F(b) - F(a)$$
   • Part 2: If $f$ is continuous on $[a, b]$, then the function $F$, defined by:
     $$F(x) = \int_a^x f(t) \,dt$$
     is continuous on $[a, b]$, differentiable on $(a, b)$, and $F'(x) = f(x)$.

3. Definite Integrals:

   • The definite integral calculates the net area under the curve of a function over an interval $[a, b]$.
   • Properties of definite integrals include:
     • Linearity: $\int_a^b [cf(x) + g(x)]\,dx = c\int_a^b f(x)\,dx + \int_a^b g(x)\,dx$
     • Additivity: $\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$
     • Reversal of limits: $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$

4. Applications:

   • Use definite integrals to solve problems related to area, volume, work, and other physical applications.
   • Understand how to apply the FTC to evaluate definite integrals efficiently.

5. Limit of Riemann Sums:

   • Recognize that the definite integral can be defined as the limit of Riemann sums as the partition of the interval becomes infinitely small.

6. Techniques of Integration:

   • Familiarize yourself with methods such as substitution, integration by parts, and recognizing standard forms, as they will assist in finding antiderivatives $F(x)$.

By mastering these key points, you'll gain a solid foundation in understanding the Fundamental Theorem of Calculus and the concept of definite integrals.

Here are the key points to learn when studying "Antiderivatives and Indefinite Integrals":

1. Definition: An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$. The indefinite integral symbol $\int f(x) \, dx$ represents the family of all antiderivatives of $f(x)$.

2. Notation: The notation for the indefinite integral is $\int f(x) \, dx = F(x) + C$, where $C$ is the constant of integration, accounting for all possible vertical shifts in the antiderivative.

3. Basic Antiderivative Rules:

   • Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
   • Constant Multiple Rule: $\int k f(x) \, dx = k \int f(x) \, dx$, where $k$ is a constant.
   • Sum/Difference Rule: $\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx$.

4. Common Antiderivatives:

   • $\int e^x \, dx = e^x + C$
   • $\int \sin(x) \, dx = -\cos(x) + C$
   • $\int \cos(x) \, dx = \sin(x) + C$
   • $\int \sec^2(x) \, dx = \tan(x) + C$

5. Techniques of Integration:

   • Substitution: Used when integrands are compositions of functions.
   • Integration by Parts: Based on the product rule, useful for products of functions.
   • Partial Fraction Decomposition: Useful for rational functions.

6. Fundamental Theorem of Calculus: Establishes the connection between differentiation and integration. It asserts that if $F(x)$ is an antiderivative of $f(x)$ on an interval [a, b], then:

   $$\int_a^b f(x) \, dx = F(b) - F(a)$$

7. Applications: Antiderivatives are used for calculating areas under curves, solving differential equations, and in physics for determining quantities like displacement from velocity.

8. Practice: Regular practice is essential for mastering antiderivatives, including solving a variety of problems and using different techniques of integration.

Understanding these key points will provide a solid foundation for further study and application of antiderivatives and indefinite integrals in calculus.