Links derivatives and integrals: integrating a rate over an interval gives total change.
Part 1: d/dx ∫ₐˣ f(t)dt = f(x). Part 2: ∫ₐᵇ f(x)dx = F(b)-F(a) for any antiderivative F. It makes integration practical and shows differentiation and integration are inverse processes.
∫₁² 3x² dx = 2³-1³ = 7.
This theorem is why a rate and an accumulation are two views of the same thing, so you can move between them to solve problems.
Links derivatives and integrals: integrating a rate over an interval gives total change.
This theorem is why a rate and an accumulation are two views of the same thing, so you can move between them to solve problems.
The derivative of the integral of a function is the function, and the integral of a derivative gives the net change between the endpoints.
Be careful. The theorem needs the function to be continuous or well-behaved over the interval, or the result can be wrong.
Typical applications include recovering position from a velocity curve, finding the change in stored energy from a power measurement and computing total fuel used from a flow-rate record.
Study "Integrals" first (topic 32 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Constants cancel in definite integrals. Applies to continuous functions. Basis of accumulation functions.
∫₁² 3x² dx = 2³-1³ = 7.
The next topic is "Integration techniques", which builds directly on this one.
It is classed as Beginner in Engineering Mathematics (topic 33 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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