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[rac{x^2}{2} - rac{x^3}{3} ight]_0^1 = rac{1}{2} - rac{1}{3} = rac{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 $x$.

Common Cross Sections

If the base is between $y = f(x)$ and $y = g(x)$, the side length is $s = f(x) - g(x)$.

Cross Sections 🎯

Base is the region between $y = \sqrt{x}$ and $y = 0$ from $x = 0$ to $x = 4$. Cross sections perpendicular to $x$-axis are squares.

Key Takeaways — Part 2

1. Volume = $\int A(x)\,dx$ where $A(x)$ = cross-section area
2. The side length of the cross section comes from the curve

Integration Applications

Part 3 of 7 — Volumes: Disk and Washer Methods

Disk Method (rotation about x-axis)

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

Washer Method

Disk & Washer 🎯

Key Takeaways — Part 3

1. Disk: $\pi r^2$ — one function
2. Washer: $\pi(R^2 - r^2)$ — two functions
3. Adjust radii when rotating about lines other than axes

Integration Applications

Part 4 of 7 — Riemann Sums and Trapezoidal Rule

Left, Right, and Midpoint Sums

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

Trapezoidal Rule

T_n = rac{Delta x}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + cdots + 2f(x_{n-1}) + f(x_n)]

Over/Underestimates

Numerical Integration 🎯

Given: $x = 0, 1, 2, 3$ with $f(0) = 1, f(1) = 3, f(2) = 2, f(3) = 5$.

Key Takeaways — Part 4

1. Trapezoidal rule averages left and right sums
2. Know which methods overestimate vs underestimate

Integration Applications

Part 5 of 7 — Rate Problems & Net Change

The Net Change Theorem

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

Common Contexts

• Water flow: $int R(t),dt$ = total water
• Population: $int P'(t),dt$ = net change in population
• Cost: $int C'(x),dx$ = total cost change
• Velocity: $int v(t),dt$ = displacement

Rate Problems 🎯

Water flows into a tank at $R(t) = 10 + \sin(t)$ gal/min.

Key Takeaways — Part 5

1. Integrating a rate gives total accumulation
2. Include units in your answer on the AP exam

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! ✅