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

Common mistakes and why they fail