MathDB

For

a>2

, let

f(t)=\frac{\sin ^ 2 at+t^2}{at\sin at},\ g(t)=\frac{\sin ^ 2 at-t^2}{at\sin at}\ \left(0<|t|<\frac{\pi}{2a}\right)

and
let

C: x^2-y^2=\frac{4}{a^2}\ \left(x\geq \frac{2}{a}\right).

Answer the questions as follows.
(1) Show that the point

(f(t),\ g(t))

lies on the curve

C

.
(2) Find the normal line of the curve

C

at the point

\left(\lim_{t\rightarrow 0} f(t),\ \lim_{t\rightarrow 0} g(t)\right).

(3) Let

V(a)

be the volume of the solid generated by a rotation of the part enclosed by the curve

C

, the nornal line found in (2) and the

x

-axis. Express

V(a)

in terms of

a

, then find

\lim_{a\to\infty} V(a)

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