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Feb 2, 2023 · The Fundamental Theorem of Calculus, Part 2 (also known as the evaluation theorem) states that if we can find an antiderivative for the integrand, then we can evaluate the definite integral by evaluating the antiderivative at the endpoints of the interval and subtracting.
Calculus. The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions).
As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas.
We use the first fundamental theorem of calculus in accordance with the chain-rule to solve this. Let \(u=x^{2}\), then \[\begin{align} \frac{d}{dx} \int_{2}^{x^2}{\frac{1}{1+t^2}\, dt} &= \frac { d }{ du } \left[ \int _{ 1 }^{ u }{ \frac { 1 }{ 1+{ t }^{ 2 } } dt} \right] \cdot \frac { du }{ dx } \\ &=\frac { 1 }{ 1+{ u }^{ 2 } } \cdot 2{ x ...
Fundamental Theorem Of Calculus: The original function lets us skip adding up a gajillion small pieces. In the upcoming lessons, we’ll work through a few famous calculus rules and applications. The real goal will be to figure out, for ourselves, how to make this happen:
