Question

Find the area of the region bounded by the parabola y = 5x^2, the tangent line to this parabola at (4, 80), and the x-axis.

Answer 1

First we have to find the tangent line:
y - y o = f ` ( x o ) * ( x - x o )
f ` ( x ) = 10 x
y - 80 = 40 ( x - 4 )
y = 40 x - 160 + 80
y = 40 x - 80
There are 2 parts of this region:
A =$\int\limits^2_0 {5 x^{2} } \, dx + \int\limits^4_2 {(5 x^{2} -40x+80)} \, dx= \\ 5 x^{3} /3 \left \{ {{2} \atop {0}} \right. +( 5 x^{3} -20 x^{2} +80x) \left \{ {{4} \atop {2}} \right.$
A = 40/3 + 320/3 - 320 + 320 - 40/3 + 80 - 160 = 320/3 - 240/3 = 80/3