Integration Applications

Integration Applications

Part 1 of 7 — Area Between Curves (Advanced)

Area Between Two Curves

$A = int_a^b |f(x) - g(x)|,dx$

When $f(x) geq g(x)$ on $[a,b]$:

$A = int_a^b [f(x) - g(x)],dx$

When Curves Cross

Split the integral at intersection points!

Worked Example

Area between $y = x^2$ and $y = x$ on $[0, 1]$:

Intersection: $x^2 = x implies x = 0, 1$.

On $[0,1]$: $x geq x^2$.

A = int_0^1 (x - x^2),dx = left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}

Area Between Curves 🎯

Key Takeaways — Part 1

1. Always determine which curve is on top
2. Find intersection points to set limits
3. Split integral if curves cross within the interval

Integration Applications

Part 2 of 7 — Cross-Sectional Volumes

Volume with Known Cross Sections

$V = int_a^b A(x),dx$

where $A(x)$ is the area of the cross section at position .

Integration Applications

Part 3 of 7 — Volumes: Disk and Washer Methods

Disk Method (rotation about x-axis)

$V = piint_a^b [f(x)]^2,dx$

Integration Applications

Part 4 of 7 — Riemann Sums and Trapezoidal Rule

Left, Right, and Midpoint Sums

$L_n = sum_{i=0}^{n-1} f(x_i),Delta x qquad R_n = sum_{i=1}^{n} f(x_i),Delta x$

Integration Applications

Part 5 of 7 — Rate Problems & Net Change

The Net Change Theorem

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

Integration Applications

Part 6 of 7 — Practice Workshop

Mixed Integration Applications 🎯

Workshop Complete!

Integration Applications — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Integration Applications — Complete! ✅