Exercise · ZW-POLAR-COORDINATES-005 · ★★☆ Intermediate
Compute \(\iint_D (x^2+y^2)\,dA\), where \(D\) is the unit disc \(x^2+y^2\le1\).
Choose one option.
Show answer
- D. \(\tfrac{\pi}{2}\)
Explanation
In polar form it is \(\int_0^{2\pi}\!\int_0^1 r^2\cdot r\,dr\,d\theta=2\pi\cdot\tfrac14=\tfrac{\pi}{2}\). Forgetting the factor \(r\) in \(dA\) would give \(\tfrac{2\pi}{3}\).
Review the concept: Polar coordinates
Polar coordinates locate a point of the plane by its distance \(r\) from the origin and the angle \(\theta\) its radius makes with the positive \(x\)-axis.
\[x=r\cos\theta,\quad y=r\sin\theta\]
More practice problems: polar coordinates
- ZW-POLAR-COORDINATES-001 Find the x-coordinate of the point with r=4 and θ=tfracπ3.
- ZW-POLAR-COORDINATES-002 The Cartesian point (-1,1) written in polar form with r>0 and 0≤θ<2π is:
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- ZW-POLAR-COORDINATES-004 Using dA=r dr dθ, compute the area of the disc of radius 2: ∫_0^2π\!∫_0^2 r dr dθ.
- ZW-POLAR-COORDINATES-006 Find the area of the half-disc 0≤ r≤3, 0≤θ≤π.
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