題目 9. (10 Points) Evaluate the triple integral ∭Ex2+y2 dV\iiint_E \sqrt{x^2+y^2}\,dV∭Ex2+y2dV, where EEE is the solid region in R3\mathbb{R}^3R3 bounded by the surface z=1z=1z=1 and z=x2+y2z=x^2+y^2z=x2+y2. 解答 解法一 用圓柱座標。區域為 0≤r≤10\le r\le10≤r≤1、r2≤z≤1r^2\le z\le1r2≤z≤1、0≤θ≤2π0\le\theta\le2\pi0≤θ≤2π，而被積函數為 rrr： ∭Ex2+y2 dV=∫02π∫01∫r21r2 dz dr dθ=4π15.\iiint_E\sqrt{x^2+y^2}\,\mathrm{d}V =\int_0^{2\pi}\int_0^1\int_{r^2}^1r^2\,\mathrm{d}z\,\mathrm{d}r\,\mathrm{d}\theta =\boxed{\frac{4\pi}{15}}.∭Ex2+y2dV=∫02π∫01∫r21r2dzdrdθ=154π.
