Applications of Integration

The central idea

A definite integral accumulates a density: if a small slice of thickness $dx$ contributes approximately $q(x)dx$, then the total is $\int_a^bq(x)dx$. Work uses $W=\int F(x)dx$; mass uses $m=\int\rho(x)dx$ for linear density; fluid force uses pressure times strip area. Bounds and units are part of the model.

Definitions, hypotheses, and notation

In pumping problems, a horizontal liquid slice usually contributes $dW=(\text{weight density})(\text{slice area})(\text{lifting distance})dy$. All three factors may vary. In hydrostatic-force problems the slice does not move; instead $dF=(\text{weight density})(\text{depth})(\text{strip width})dy$. Distinguishing work from force prevents using the wrong distance factor.

A constant-density shortcut is legitimate only when both density and travel distance are constant. Otherwise, integration weights each location correctly. A quick bounding estimate—total amount times minimum and maximum relevant factor—should contain the final result.

Conceptual meaning

The unifying move is to model one representative thin piece, not to memorize unrelated formulas. The integral is a limit of the sum of those piece contributions. Different applications change the density factor but preserve this architecture.

A dependable method and decision rule

1. Draw the physical setup and choose a coordinate.
2. Describe a slice at position $x$ with thickness $dx$.
3. Build the slice contribution from local density, distance, force, or area.
4. Set bounds that cover each physical piece exactly once.
5. Integrate and verify units, sign, and plausible size.

Fully worked example

Graphical or geometric meaning

Imagine coloring each cable element by how far it travels. Elements near the top contribute little work; elements near the bottom contribute most. The graph of $2x$ against depth has triangular area 100, matching the integral.

Common mistakes and why they fail

Verification and reasonableness checks

• Derive integrand units and confirm integration leaves target units.
• Compare with upper and lower estimates from minimum and maximum slice density.
• Check that bounds correspond to the chosen coordinate, not a different diagram label.

Build the slice before writing the integral

Every application begins with a small contribution whose units reveal the model. Force integrated over distance gives work; density times a small length, area, or volume gives mass; and cross-sectional area times thickness gives volume. Draw one representative slice, label its location and thickness, and express all dimensions in one variable. Bounds must describe the physical region, not merely convenient algebra. A dimensional check is powerful: if a proposed mass integral ends with length units, a density factor or geometric measure is missing. Symmetry may justify doubling half a region only when both geometry and density share it. After evaluation, compare with an estimate such as average density times total size to catch an implausible scale or sign.

Practice

1. A constant 3-N force moves an object 5 m. Find work.
2. A rod on $[0,2]$ has density $1+x$. Find mass.
3. Why is hydrostatic pressure deeper in a fluid larger?

Connections and next steps

• Definite Integral
• Area Between Curves
• Volumes by Cylindrical Shells