Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-013 · ★☆☆ Basic
Evaluate the area under one arch of the sine: \(\displaystyle\int_0^{\pi}\sin x\,dx\).
Choose one option.
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- B. \(2\)
Explanation
An antiderivative is \(-\cos x\); the integral is \(-\cos\pi+\cos 0=1+1=2\).
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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