Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-007 · ★★☆ Intermediate
Compute \(\displaystyle\int_{0}^{4} \sqrt{x}\,dx\).
Choose one option.
Show answer
- B. \(\frac{16}{3}\)
Explanation
An antiderivative is \(F(x)=\frac{2 x^{\frac{3}{2}}}{3}\). By the fundamental theorem, the integral is \(F(4)-F(0)=\frac{16}{3}\).
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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