Integral from a to a of f(x) with respect to x
The integral from a to a of f(x) with respect to x is denoted as ∫[a to a] f(x) dx
The integral from a to a of f(x) with respect to x is denoted as ∫[a to a] f(x) dx.
When integrating over this range, the lower limit (a) is the same as the upper limit (a). This means that the interval over which the integration is being performed has zero width.
Mathematically, the integral from a to a of f(x) is always zero, regardless of the function f(x). This can be proven using the fundamental theorem of calculus.
According to the fundamental theorem of calculus, if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b can be calculated as the difference between the antiderivatives evaluated at the upper and lower bounds. In this case, the antiderivative of f(x) is F(x), and evaluating it at both a and a gives F(a) – F(a) = 0.
To summarize, the integral from a to a of any function f(x) is always equal to zero.
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