Mathematics

Problem

Let

T

be the region in the first quadrant that is bounded by the

x

–axis and the curve

y=e^{-x^{2}}

for

x\ge 0

. (a) Write, but do not yet evaluate, an integral expression that gives the exact area of

T

. Be sure to include any limit notation that is required. (b) Evaluate the integral from part (a) to show that the area of

T

equals

\dfrac{\sqrt{\pi}}{2}

. (You may use the fact that

\displaystyle \int_{0}^{\infty} e^{-u^{2}}\,du=\dfrac{\sqrt{\pi}}{2}

.) (c) The region

T

is revolved about the

x

–axis to generate a solid of revolution. Write, and then evaluate, an integral that gives the exact volume of this solid.