Evaluate area under curves and work done by variable forces.
Definite integrals ∫ₐᵇ f dx give net signed area. The area between curves is ∫(top-bottom)dx. Average value is (1/(b-a))∫f dx, and the RMS value of a waveform uses the square root of the average of f².
The RMS of A sin ωt is A/√2 because the average of sin² is 1/2.
Definite integrals give areas and work with variable forces, which are routine quantities in mechanics and fluid problems.
Evaluate area under curves and work done by variable forces.
Definite integrals give areas and work with variable forces, which are routine quantities in mechanics and fluid problems.
Integrate the force with respect to position over the displacement, since the work done in a small step is force times the step.
A definite integral gives a signed area. Regions below the axis count negatively, so take absolute values when you need a physical area.
Typical applications include computing the work to compress a spring from its rest length, finding the area under an efficiency curve and calculating the total load from a non-uniform pressure distribution.
Study "Integration techniques" first (topic 34 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Split at crossings for absolute area. Symmetry can simplify. Units are f-units times x-units.
The RMS of A sin ωt is A/√2 because the average of sin² is 1/2.
The next topic is "Volumes of revolution", which builds directly on this one.
It is classed as Beginner in Engineering Mathematics (topic 35 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
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