Integrals -zambak-

Zambak books focus heavily on these three methods:

1. Substitution Method (Change of Variable) Used for composite functions. Let $u = g(x)$, then $du = g'(x)dx$. $$ \int f(g(x)) \cdot g'(x) , dx = \int f(u) , du $$

2. Integration by Parts Used for products of functions (e.g., $x \cdot e^x$ or $x \cdot \ln x$). Formula: $$ \int u , dv = u \cdot v - \int v , du $$ (Typical mnemonic in Zambak books for choosing $u$: LIATES - Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential). Integrals -Zambak-

3. Partial Fractions (Simple) Used when integrating rational functions $\fracP(x)Q(x)$. You decompose the fraction into simpler terms. Example: $$ \frac1(x-1)(x+2) = \fracAx-1 + \fracBx+2 $$

| Application | Integral Form | |---|---| | Area under curve | ( \int_a^b f(x) , dx ) | | Area between curves | ( \int_a^b [f(x) - g(x)] , dx ) | | Volume (disk method) | ( \pi \int_a^b [R(x)]^2 dx ) | | Work by variable force | ( \int_x_1^x_2 F(x) , dx ) | | Average value | ( \frac1b-a \int_a^b f(x) dx ) | | Displacement from velocity | ( \int_t_1^t_2 v(t) dt ) | Zambak books focus heavily on these three methods: 1

Zambak Example 6 (Area):
Find area between ( y = x^2 ) and ( y = x ) from ( x=0 ) to ( x=1 ).

Solution:
[ \int_0^1 (x - x^2) dx = \left[ \fracx^22 - \fracx^33 \right]_0^1 = \frac12 - \frac13 = \frac16 ] | Application | Integral Form | |---|---| |

Every theorem in the Zambak book is accompanied by high-quality diagrams. For instance, when explaining the Washer Method for volume, the book shows a cross-section of the solid, the radius functions, and the resulting integral all in one cohesive graphic. This visual scaffolding is invaluable for kinesthetic and visual learners.