Today's calculation of Integral 559
Source: 2010 Waseda University entrance exam/Science and Technology
February 17, 2010
calculusintegrationgeometrygeometric transformationrotationtrigonometrycalculus computations
Problem Statement
In space, consider two points P(1,\ 0,\ 1),\ Q(\minus{}1,\ 1,\ 0). Let be the surface generated by rotation the line segment about axis. Answer the following questions.
(1) Find the volume of the solid bounded by the surface and two planes x\equal{}1 and x\equal{}\minus{}1.
(2) Find the cross-section of the solid in (1) by the plane y\equal{}0 to sketch the figure on the palne y\equal{}0.
(3) Evaluate the definite integral \int_0^1 \sqrt{t^2\plus{}1}\ dt by substitution t\equal{}\frac{e^s\minus{}e^{\minus{}s}}{2}.
Then use this to find the area of (2).