MathDB

Today's calculation of Integral 123

Source: Tokyo Univesity entrance exam/Science 1989

July 11, 2006

calculusintegrationtrigonometrylimitgeometryfunction3D geometry

Problem Statement

Let

f(x)=\pi x^{2}\sin \pi x^{2}.

Prove that the volume

V

formed by the revolution of the figure surrounded by the part

0\leq x\leq 1

of the graph of

y=f(x)

and

x

axis about

y

axis is can be given as

2\pi \int_{0}^{1}xf(x)\ dx

then find the value of

V.