A thin flat rigid plate occupies the diamond-shaped region in the -plane

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist

Problem

A thin flat rigid plate occupies the diamond-shaped region

R = \{(x,y): |x+y|\le1\,\mathrm{m},\ |x-y|\le1\,\mathrm{m}\}

in the

xy

-plane. Its second moment of area about the axis through the origin perpendicular to the plate is

I=\iint_R (x^2+y^2)\,\mathrm{d}A

. ((a)) Introduce

u=x+y

v=x-y

, and check the hypotheses of the change-of-variables theorem for this map: bijectivity, continuous differentiability, and a nowhere-vanishing Jacobian, over

R

. ((b)) Evaluate

I

by changing variables to

u,v

Browse the problem library