Question

Find the area of the region.
y=8x , y=5x^2
CHOICE C у 14 12 10 8 6 4 2. - X 0.5 1.0 1.5

Answer 1

256/75 or about 3.143

Explanation:

Find intersection points

`8x=5x^2\\8x-5x^2=0\\x(8-5x)=0\\x=0,\,x=(8)/(5)`

Set up integral and evaluate

`\displaystyle A=\int^b_a(\text{Upper Function}-\text{Lower Function})dx\\\\A=\int^(8)/(5)_0(8x-5x^2)dx\\\\A=4x^2-(5)/(3)x^3\biggr|^(8)/(5)_0\\\\A=4\biggr((8)/(5)\biggr)^2-(5)/(3)\biggr((8)/(5)\biggr)^3\\\\A=4\biggr((64)/(25)\biggr)-(5)/(3)\biggr((512)/(125)\biggr)\\\\A=(256)/(25)-(2560)/(375)\\\\A=(3840)/(375)-(2560)/(375)\\\\A=(1280)/(375)\\\\A=(256)/(75)=3.41\overline{3}`

I've attached a graph of the area between the two curves in case it helps you understand better!