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michonamona
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Homework Statement
F(x) = [tex]\int^{x}_{0}f(t)dt[/tex]
Then F'(x) = f(x)
what is f'(x)? is this equivalent to f(t)?
Thanks for your help
M
michonamona said:Thanks for the reply.
so f '(x) IS indeed f(t)? the very same f(t) in F(x)?
michonamona said:I'm sorry, I don't think my notations are clear. I understand that big F'(x) = f(x), what I'm concerned with is whether small f '(x) is f(t).
so can we write
F(x) = [tex]\int^{x}_{0}f'(x)dt[/tex] = [tex]\int^{x}_{0}f(t)dt[/tex]
A definite integral is a mathematical concept used to find the area under a curve between two points on a graph. It is represented by the symbol ∫ and is calculated by taking the limit of a sum of infinitely small rectangles under the curve.
A definite integral has specific limits of integration, whereas an indefinite integral does not. This means that a definite integral will give a numerical value, while an indefinite integral gives a function.
Definite integrals are used in a variety of fields, such as physics, engineering, and economics, to calculate quantities such as distance, volume, and work. They are also used in the process of finding antiderivatives.
Yes, definite integrals can be negative. The sign of a definite integral depends on the direction of the curve and the limits of integration. A negative definite integral represents a region below the x-axis on a graph.
To solve a definite integral, you can use the fundamental theorem of calculus or integration techniques such as substitution, integration by parts, or partial fractions. It is also important to correctly identify the limits of integration and any special properties of the integrand.