Answer :
The second fundamental theorem of calculus states that if f is continuous on [a,b] and F is an antiderivative of f on the same interval, then:
[tex]\int\limits^b_a[/tex]f(x) dx = F(b) - F(a)
The second fundamental theorem of calculus states that if f is continuous on [a,b] and F is an antiderivative of f on the same interval, then:
[tex]\int\limits^b_a[/tex]f(x) dx = F(b) - F(a)
It uses the mean value theorem of integration and the limit of an infinite Riemann summation. But I tried coming up with a proof and it was barely two lines. Here it goes:
Since F is an antiderivative of f, we have dFdx=f(x). Multiplying both sides by dx, we obtain dF=f(x)dx. Now, dF is just the small change in F and f(x)dx represents the infinitesimal area bounded by the curve and the x axis. So integrating both sides, we arrive at the required result.
To learn more about fundamental theorem:
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