Exercise · ZW-POLAR-COORDINATES-009 · ★★★ Advanced
Integrating \(e^{-(x^2+y^2)}\) over the disc of radius \(a\) and letting \(a\to\infty\), the value of \(\iint_{\mathbb{R}^2}e^{-(x^2+y^2)}\,dA\) is:
Choose one option.
Show answer
- B. \(\pi\)
Explanation
\(\int_0^{2\pi}\!\int_0^a e^{-r^2}r\,dr\,d\theta=\pi(1-e^{-a^2})\to\pi\). Since this is the square of \(\int_{-\infty}^{\infty}e^{-x^2}dx\), that integral equals \(\sqrt\pi\).
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:
- ZW-POLAR-COORDINATES-003 The circle x^2+y^2=9 is written in polar coordinates as:
- 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-005 Compute iint_D (x^2+y^2) dA, where D is the unit disc x^2+y^2≤1.
- ZW-POLAR-COORDINATES-006 Find the area of the half-disc 0≤ r≤3, 0≤θ≤π.
- ZW-POLAR-COORDINATES-007 The polar curve r=2cos θ is:
- ZW-POLAR-COORDINATES-008 Compute iint_D x dA, with D the quarter disc of radius 2 in the first quadrant.