Question

Find the area of the region bounded by the parabola y=5x2y=5x2, the tangent line to this parabola at (2,20)(2,20) and the xx axis.

Answer 1

1. Find the equation of tangent line at point (2,20).

$y'=(5x^2)'=5\cdot 2x=10x,\\ \\y'(2)=10\cdot 2=20.$

The equation of the tangent line is

$y=20(x-2)+20,\\ \\y=20x-20.$

2. Express x:

$y=20x-20\Rightarrow x=\dfrac{y}{20}+1;\\ \\y=5x^2\Rightarrow x=\sqrt{\dfrac{y}{5}}.$

3. Find the area of bounded region:

$A=\int\limits^{20}_0 {\left(\dfrac{y}{20}+1-\sqrt{\dfrac{y}{5}} } \right)\, dy=\left(\dfrac{y^2}{40}+y-\dfrac{2\sqrt{y^3} }{3\sqrt{5} } \right)\big|^{20}_0=$

$=\dfrac{400}{40}+20-\dfrac{2\sqrt{20^3} }{3\sqrt{5} }=30-\dfrac{80}{3}=\dfrac{10}{3}\ sq. un.$

Answer: $\dfrac{10}{3}\ sq. un.$