The Fundamental Theorem of Calculus

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Integration by Parts

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Integrals of Trig Functions

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Differential Equations

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Infinite Sequences

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Infinite Series

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Integral Test

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Comparison Tests

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Convergence of Series with Negative Terms

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Alternating Series and the AS Test
Absolute Convergence
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The Ratio and Root Tests

The Ratio Test
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Strategies for testing Series

List of Major Convergence Tests
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Power Series

Radius and Interval of Convergence
Finding the Interval of Convergence
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Representing Functions as Power Series

Functions as Power Series
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Taylor and Maclaurin Series

The Formula for Taylor Series
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Applications of Taylor Polynomials

What are Taylor Polynomials?
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Partial Derivatives

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Differentials and Taylor Expansions

Multiple Integrals

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What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

One Variable at the Time
Fubini's Theorem
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Double Integrals over General Regions

Type I and Type II regions
Examples
Order of Integration
Area and Volume Revisited

The FTC and the Chain Rule

By combining the chain rule with the (second) Fundamental Theorem of Calculus, we can solve hard problems involving derivatives of integrals.

Example: Compute

${\displaystyle\frac{d}{dx} \int_1^{x^2} \tan^{-1}(s)\, ds.}$

Solution: Let

$F(x)$ be the anti-derivative of

$\tan^{-1}(x)$ . Finding a formula for

$F(x)$ is hard, but we don't actually need the formula!

$\int_1^{x^2} \tan^{-1}(s) \,ds = F\left(x^2\right) - F(1),$ so

$\frac{d}{dx} \int_1^{x^2} \tan^{-1}(s)\,ds = 2x F'\left(x^2\right) = 2x f\left(x^2\right) = 2x \tan^{-1}\left(x^2\right).$

This method generalizes:

$f$ is a continuous function and

$g$ and

$h$ are differentiable functions, then

$\frac{d}{dx} \int_{g(x)}^{h(x)} f(s)\, ds = \frac{d}{dx} \Big[F\left(h(x)\right) - F\left(g(x)\right)\Big]$

$\hspace{3cm}\quad\quad\quad= F'\left(h(x)\right) h'(x) - F'\left(g(x)\right) g'(x)$

$\hspace{3cm}\quad\quad = f\left(h(x)\right) h'(x) - f\left(g(x)\right) g'(x).$