Exercise · ZW-DEFINITE-INTEGRAL-001 · ★★☆ Intermediate
What is \(\displaystyle\int_{-1}^{1} x\,dx\)?
Choose one option.
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- A. 0
Explanation
The function x is odd: the area above the axis (from 0 to 1) is 1/2 and the area below it (from −1 to 0) is also 1/2 with negative sign. The definite integral is a signed net area and gives 0.
Review the concept: Definite integral
The definite integral of \(f\) on \([a,b]\) is the limit of the Riemann sums as \(n\to\infty\). It represents the signed net area between the graph of \(f\) and the \(x\)-axis, from \(x=a\) to \(x=b\).
\[\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^{*})\,\Delta x\]
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