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=(y)/(20)+1;\\ \\y=5x^2\Rightarrow x=\sqrt{(y)/(5)}.`

3. Find the area of bounded region:

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

`=(400)/(40)+20-(2√(20^3) )/(3√(5) )=30-(80)/(3)=(10)/(3)\ sq. un.`

Answer:
`(10)/(3)\ sq. un.`