Sketch the region enclosed by the curve and calculate it's area?

Question

Sketch the region enclosed by the curve and calculate it's area:

#y = 3x-x^2, y=0#

422 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-25T04:52:15

See below.

enter image source here

First we need to find the roots to #y=3x-x^2#, as these will be the upper and lower bounds of the area we seek.

#3x-x^2=0#

#x(3-x)=0=>x=0 and x=3#

We now need the integral:

#int_0^3(3x-x^2) dx=[3/2x^2-1/3x^3]_0^3#

#A=[3/2(3)^2-1/3(3)^3]-[3/2(0)^3-1/3(0)^3]#

#A=[9/2]-[0]#

#Area = 9/2=4.5# units squared.