Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-008 · ★☆☆ Basic
What does the second part of the fundamental theorem of calculus state?
Choose one option.
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- A. \(\int_a^b f(x)\,dx=F(b)-F(a)\), with \(F\) an antiderivative of \(f\)
Explanation
The second part says that if \(F\) is an antiderivative of a continuous \(f\), the integral of \(f\) from \(a\) to \(b\) equals \(F(b)-F(a)\).
Review the concept: Fundamental theorem of calculus
It links derivatives and integrals: a definite integral equals an antiderivative evaluated at the endpoints, and differentiating an integral with variable upper limit returns the integrand.
\[\begin{gathered}\int_a^b f(x)\,dx=F(b)-F(a)\\ \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\end{gathered}\]
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- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-007 Compute ∫_0^4 √x dx.
- ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-009 By the fundamental theorem, what is (d)/(dx)∫_0^x^2 sin (t) dt?
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