MathDB

768

Part of 2011 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 768

Source:

11/17/2011
Let rr be a real such that 0<r≤10<r\leq 1. Denote by V(r)V(r) the volume of the solid formed by all points of (x, y, z)(x,\ y,\ z) satisfying x2+y2+z2≤1, x2+y2≤r2x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2 in xyzxyz-space.
(1) Find V(r)V(r).
(2) Find lim⁡r→1−0V(1)−V(r)(1−r)32.\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.
(3) Find lim⁡r→+0V(r)r2.\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.
calculusintegrationlimitgeometry3D geometryspheresymmetry