Exercise · ZW-DOUBLE-INTEGRAL-008 · ★★☆ Intermediate
Compute \(\iint_D x\,dA\) over the unit disk \(D:\ x^2+y^2\le1\).
Choose one option.
Show answer
- A. \(0\)
Explanation
The region is symmetric about the \(y\)-axis and \(f(x,y)=x\) is odd in \(x\): the contributions from \(x>0\) and \(x<0\) cancel. In polar form it becomes \(\tfrac13\int_0^{2\pi}\cos\theta\,d\theta=0\).
Review the concept: Double integral
The double integral of a function \(f\) over a region \(D\) of the plane is the limit of the sums of the values of \(f\) times the area \(\Delta A\) of ever finer cells covering \(D\).
\[\iint_D f(x,y)\,dA=\lim_{\max\Delta A_i\to0}\sum_i f(x_i^*,y_i^*)\,\Delta A_i\]
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