Exercise · ZW-DEFINITE-INTEGRAL-010 · ★★☆ Intermediate
If \(\int_0^3 f(x)\,dx=10\) and \(\int_0^1 f(x)\,dx=4\), what is \(\int_1^3 f(x)\,dx\)?
Choose one option.
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- D. 6
Explanation
By additivity, the integral from 0 to 3 is the integral from 0 to 1 plus the one from 1 to 3; so 10 = 4 + X, hence X = 6.
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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