Lesson 5: Calculus 2

$$ fr 12 to 2 fx dx div z + srt 2
$$ fourier form with n

1. Integration Techniques

• Integration by Parts: Used for integrals of products (e.g., ∫x*cos(x) dx). Formula: ∫u dv = uv - ∫v du.
• Trigonometric Substitution: Used for integrals involving √(a²-x²), √(a²+x²), √(x²-a²).

2. Applications of Integration

• Area Between Curves: Finding the area enclosed between two functions, f(x) and g(x), from x=a to x=b using ∫[f(x) - g(x)] dx.
• Volumes of Revolution (Disk/Washer/Shell Methods): Calculating the volume of solids generated by revolving a volume of solids generated by revolving a region around an axis.

3. Sequences and Series

• Convergence Tests (Integral Test, Comparison Test, Ratio Test, Root Test): Determining if an infinite series converges or diverges.
• Taylor and Maclaurin Series: Representing functions as infinite sums (power series). Example: Finding the area between y=x² and y=x from x=0 to x=1. Area = ∫[x - x²] dx from 0 to 1 = [x²/2 - x³/3] from 0 to 1 = (1/2 - 1/3) - (0 - 0) = 1/6.