For any real t,
`x=2+(e^(t)+e^(-1))/(2),y=2+(e^(t)-e^(-t))/(2)` is a point on the hyperbola `x^(2)-y^(2)-4x+4y-1=0`. Find the area bounded by the hyperbola and the lines joining the center to the points corresponding to `t_(1) and -t_(1)`.

Explanation

It is a point on hyperbola `x^(2)-y^(2)=1`.
Area `(PQRP) = 2 underset(1)overset(e^(t_(1))+e^(-t_(1)))overset(2)(int)ydx = 2 underset(1)overset(e^(t_(1))+e^(-t_(1)))overset(2)(int)sqrt(x^(2)-1)dx`
`= 2[x/2sqrt(x^(2)-1)-1/2l(x+sqrt(x^(2)-1))]_(1)^(e^(t_(1))+e^(-t_(1))) = (e^(2t_(1))-e^(2t_(1)))/(4)-t_(1)`
Area of `DeltaOPQ = 2 xx 1/2((e^(t_(1))+e^(t_(1)))/(2))((e^(t_(1))-e^(t_(1)))/(2)) = (e^(2t_(1))-e^(-2t_(1)))/(4)`
`:.` Required area = area `DeltaOPQ` - area `(PQRP) = t_(1)`.
(b) If `g(y) le 0` for `in [c,d]` then area bounded by curve `x = g(y)` and y-axis between abscissa `y = c`, and `-underset(y=c)overset(d)intg(y)dy`