Exam Cram: Applications of Integration

Topic Weighting

[!IMPORTANT] This module typically accounts for 20–25% of a standard Calculus II exam. It is highly cumulative, requiring mastery of substitution ($u-substitution) and fundamental integration rules.

Topic	Frequency	Difficulty
Volumes of Revolution (Disk/Washer/Shell)	High	★★★★☆
Area Between Curves	High	★★☆☆☆
Work and Physical Applications	Medium	★★★☆☆
Arc Length & Surface Area	Low/Medium	★★★☆☆
Moments and Centers of Mass	Medium	★★★★☆

Key Concepts Summary

Area Between Curves: The integral of the "top" function minus the "bottom" function. If functions intersect, you must split the integral at the intersection points.
Volumes by Slicing:
- Disk Method: Used when there is no "hole" in the solid.
- Washer Method: Used when the region is bounded by two functions, creating a hollow center.
- Cylindrical Shells: Often easier when revolving around an axis parallel to the dependent variable's axis.
Physical Applications:
- Work: The accumulation of force over a distance (W = \int F(x) dx). Common for springs (Hooke's Law) and pumping liquids.
- Hydrostatic Force: Force exerted by a fluid on a submerged plate; depends on depth and area.
Centroids: The geometric center of a region. For a thin plate of constant density, it is the point (\bar{x}, \bar{y})$ where the plate would balance.

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Common Pitfalls

Incorrect Radius in Washers: Students often use $(R - r)^2 instead of the correct R^2 - r^2. Don't subtract radii before squaring.
Integration Limits: Forgetting to change x $-limits to$ y-limits when integrating with respect to y.
Shell vs. Washer Confusion: Using 2\pi $for washers or$ \pi for shells. Remember: Shells = 2\pi $(circumference), Washers =$ \pi (area).
Units of Work: Confusing mass and weight in US Customary units. Weight is a force (lb $), but mass in SI ($ kg) must be multiplied by g = 9.8 $to get Newtons ($ N).

Mnemonics / Memory Triggers

Area: "Top Minus Bottom" (TMB) or "Right Minus Left" (RML).
Disk/Washer: \pi \int (R^2 - r^2) , d(\text{axis}). Think of it as "Pie on the Plate" (Area of a circle).
Shells: 2\pi \int rh , d(\text{radius}). Think of it as "Two Pies in a Shell" (Circumference of a circle).
Arc Length Formula: Look for the "1" and the "Prime". L = \int \sqrt{1 + (f')^2}.

Formula / Equation Sheet

Application	Formula (x-axis / dx)	Notes
Area	A = \int_{a}^{b} [f(x) - g(x)] , dx$	$f(x) \ge g(x)
Disk Volume	V = \pi \int_{a}^{b} [R(x)]^2 , dx	No inner radius
Washer Volume	V = \pi \int_{a}^{b} ([R(x)]^2 - [r(x)]^2) , dx$	$R$ is outer, $r is inner
Shell Volume	V = 2\pi \int_{a}^{b} x f(x) , dx	Rotation about y-axis
Arc Length	L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} , dx	Function must be smooth
Work (Spring)	W = \int_{a}^{b} kx , dx$	$k = spring constant
Moment (M_y$)	$M_y = \rho \int_{a}^{b} x [f(x) - g(x)] \, dx$	Distance to y-axis is $x$

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Practice Set

Compound Region: Find the area bounded by $y = x^2$ $y = x^{2}$ and $y = 2x - x^2.
- Tip: Set them equal to find bounds (x=0, x=1).
Washer Method: Revolve the region bounded by y = \sqrt{x} $and$ $an d$ y = x^2 about the x-axis.
- Answer Setup: V = \pi \int_{0}^{1} ((\sqrt{x})^2 - (x^2)^2) , dx.
Shell Method: Revolve the region bounded by y = e^{-x^2}, y=0, x=0, x=1 about the y-axis.
- Tip: This requires u-substitution after setting up the 2\pi x f(x)$ integral.
Work (Pumping): A rectangular tank (10ft long, 5ft wide, 6ft deep) is full of water ($62.4 , lb/ft^3$). Find the work to pump all water over the top edge.
- Recall: $W = \int (Weight Density) \cdot (Area) \cdot (Distance to lift) \, dy$ .
Centroid: Find the center of mass of a semicircular plate of radius $r centered at the origin.
- Symmetry Tip: By symmetry, \bar{x} = 0. You only need to solve for \bar{y}$.