Question 1177659
Sketch the graph of the function {{{y = 20x-x^2}}}


{{{graph( 600, 600, -10, 35, -10, 110, 20x-x^2) }}}


divide the area in {{{10}}} rectangles 



In order to figure the width of each rectangle we can use the following formula:

 Δ {{{x=(b-a)/n}}}

in this case {{{a=0}}},{{{ b=20}}} and {{{n=10}}} so we get:

Δ {{{x=(20-0)/10=2}}}

so {{{each}}} rectangle must have a {{{width}}} of {{{2 }}}units

We can now calculate the height of each rectangle. So we figure the y-value of each corner of the rectangles. We get the following heights:

{{{h[1]=36}}} ..........{{{x=2 }}}units->{{{y = 20*2-2^2=36}}}

{{{h[2]=64}}}..............{{{x=4 }}}units->{{{y = 20*4-4^2=64}}}

{{{h[3]=84}}}.............{{{x=6 }}}units->{{{y = 20*6-6^2=84}}}...and so on

{{{h[4]= 96}}}

{{{h[5]=100}}}

{{{h[6]=96}}}

{{{h[7]=84}}}

{{{h[8]=64}}}

{{{h[9]=36}}}

{{{h[10]=0}}}

so now we can use the following formula to find the area under the graph. Basically what the formula does is add the areas of the rectangles:

{{{A= 2h[1]+2h[2]+2h[3]+2h[4]+2h[5]+2h[6]+2h[7]+2h[8]+2h[9]+2h[10]}}}

{{{A=2(36+64+84+96+100+96+84+64+36+0)}}}

{{{A=1320}}} -> approximate the area under the curve in the interval [{{{0}}}, {{{20}}}]