Posted by Ryan on .
Find the area of the region bounded by the parabola y = 4x^2, the tangent line to this parabola at (4, 64), and the xaxis.

calculous 
Steve,
First, find the equation of the tangent line:
y = 4x^2
y' = 8x
slope at (4,64) = 32
(y64)/(x4) = 32
y = 32x  64
32x64 crosses the xaxis at x=2
So, we need to break the area up into two parts.
Area between the curve and y=0 on [0,2]
Area between curve and tangent line on [2,4]
Area = Int(4x^2 dx)[0,2] + Int(4x^2  (32x64))[2,4]
= (4/3 x^3)[0,2] + (4/3 x^3  16x^2 + 64x)[2,4]
= [4/3 * 8] + [4/3 * 64  16*16 + 64*4]  [4/3 * 8  16*4 + 64*2]
= 64/3