MathDB

Today's calculation of Integral 791

Source: 2012 Tokyo University entrance exam/Science, problem 3

February 25, 2012

calculusintegrationalgebrafunctiondomainanalytic geometryinequalities

Problem Statement

Let

S

be the domain in the coordinate plane determined by two inequalities:

y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.

Denote by

V_1

the volume of the solid by a rotation of

S

about the

x

-axis and by

V_2

, by a rotation of

S

about the

y

-axis.
(1) Find the values of

V_1,\ V_2

.
(2) Compare the size of the value of

\frac{V_2}{V_1}

and 1.