Find the area of the region bounded by the hyperbola 25x^2 − 4y^2 = 100 and the line x = 3.

First graph the hyperbola. The vertices of hyperbola are at x= +-2.
The region bounded is from x=2 to x=3 on right side of hyperbola.
Notice we can use symmetry here. The hyperbola is reflected about the x-axis so the area below equals area above. This means we only have to worry about finding area of region from x=2 to x=3 above x-axis, then double it to get total area.

Next we need to set up an integral to find top area. We need a function of x. Solve hyperbola for y.


Area = 2*Integral from 2 to 3


Now lets focus on finding anti-derivative, then go back and apply limits to get Area.


Step 1: Use trig substitution.  
x = 2sec(w)
dx = 2sec(w)tan(w) dw



Step 2: Use integration by parts
u = 4tan(w)               dv = sec(w) tan(w)
du = 4sec^2             v = sec(w)



This gives same integral on both sides, it can be simplified to:


Step 3: Sub back in x.
x = 2sec ----> sec = x/2
tan^2 = sec^2 - 1   -----> tan = sqrt(x^2 -4)/2



Finally, apply limits and find Area: