In the process of studying calculus, you quickly realize that there are two major themes: differentiation and integration. Differential calculus helps us explain slope and tangents whereas integral calculus helps us solve problems involving area.
Total Change Theorem
And at first glance it may seem that these two ideas are disjointed, they are in fact intrinsically connected as inverse processes. As it happens, the Fundamental Theorem of Calculus, or FTC, displays this inverse relationship beautifully.
The fundamental theorem of calculus is actually divided into two parts: Fundamental Theorem of Calculus and the Second Fundamental Theorem of Calculus:
- Fundamental Theorem of Calculus Part 1 (FTC 1), pertains to definite integrals and enables us to easily find numerical values for the area under a curve.
- Fundamental Theorem of Calculus Part 2 (FTC 2), enables us to take the derivative of an integral and nicely demonstrates how the function and its derivative are forever linked, as Wikipedia asserts.
Properties of the Definite Integral
These two critical forms of the Fundamental Theorem of Calculus, allows us to make some remarkable connections between the geometric and analytical components of indefinite and definite integrals.
We will also see how our Integration Rules and Properties.
The properties of the definite integral, along with the Fundamental Theorem, help us when we are not explicitly given a function, or when we are only given a velocity graph.
The fundamental theorem of calculus is not only easy to understand and implement but is considered by many to be one of the greatest achievements in all of mathematics.
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