Description

With a firm foundation of the derivative, we next explore the idea of the antiderivatives which leads to integration. In this topic, we explore finding antiderivatives, evaluating indefinite integrals and using u-substitutions; exploring sigma notation; finding areas under graphs; exploring the Fundamental Theorem of Calculus by exploring definate integrals and the definition of a definite integral. Further, we explore properties of the definite integrals and explore approximate integration using the midpoint, trapezoidal and Simpson's rules.

Sections of this topic include:

1. Antiderivatives

   Like all functions in mathematics, antiderivatives are the opposite operation of derivatives. We refer to an antiderivative as an integral which is finding some function whose derivative becomes the current function. Groups within this section include: evaluating indefinate integrals; verifying anti-derivatives; performing derivatives and anti-derivatives; soving differential equations; solving various problems; solving differential equations using separation of variables; solving differential equations with initial conditions.

2. Indefinite Integrals and the u-Substitution

   Many times to integrate an expression, we need to use a u-Substitution. Groups of this section include: Evaluating indefinite integrals using the u-substitution; solving differential equations using the u-substitution.

3. Sigma Notation

   To explore the relationship between anti-derivatives and areas under curves, we first explore sigma notation and it's properties. Groups within this section include: expanding sigma notation sums; writing sums using sigma notation; finding sums using sigma notation formulas; determining if two sigmas are equal; writing decimals as a sum with sigma notation; and finding sums by expanding sigma notation.

4. Area Under a Graph

   We now develop an extensive formula to find areas under graphs using limits, and sigma notations. Groups within this section include: finding the area under a graph by definition; and determining a region from an area formula.

5. Definite Integral; Definition of a Definite Integral

   We now define a definite integral using Riemann Sums. Groups in this section include: finding riemann sums; using a regular partition to evaluate a definite integral; using a regular partition to evaluate a constant function.

6. Properties of the Definite Integral

   In this section we explore properties of the definite integral. The group in this section is called: Using properties of the definite integral.

7. Fundamental Theorem of Calculus

   We are now ready to investigate the Fundamental Theorem of Calculus (which in part ties the anti-derivative and definite integrals to areas under curves). Groups within this section include: using the Fundamental Theorem of Calculus - derivative form; and evaluating definite integrals using the Fundamental Theorem of Calculus.

8. Definite Integral; Definition of a Definite Integral

   We conclude this topic with using approximation techniques known as The Midpoint Rule, The Trapezoidal Rule, and Simpson's Rule to approximate the area under a function. Groups within this section include: Finding an approximation using the Midpoint Rule; and finding an approximating using the Trapezoidal Rule.