The figure above shows the graphs of the line x = (5/3) y and the curve C given by x = sqrt( 1+y*y). Let S be the shaded region bounded by the two graphs and the x-axis. The line and the curve intersect at point P.

(a) Find the coordinates of the point P and the value of dx/dy for the curve C at point P.

-Find the point of intersection

(1.25, .75)

- Find the slope of the curve at (1.25, .75)

(b) Set up and evaluate an integral expression with respect to y that gives the area of S.

A =

(c) Curve C is a part of the curve . Show that can be written as the polar equation .

x = r * cos() x² - y² = 1
y = r * sin() (r * cos())² - (r * cos())^{2 = 1}
r²cos()² - r²sin()² = 1
r²( cos()² - sin()² ) = 1

(d) Use the polar equation in part (c) to set up an integral with respect to the polar angle THETA that represents the area of S