Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-016 · ★★☆ Intermediate
Compute \(\displaystyle\int_{-1}^{2}x^2\,dx\).
Choose one option.
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- D. \(3\)
Explanation
With antiderivative \(x^3/3\): \(\tfrac83-\tfrac{-1}{3}=3\). A common slip is evaluating at \(-1\) as \(\tfrac13\) instead of \(-\tfrac13\).
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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