How do you find the area between the curves #x+3y=21# and #x+7=y^2#?

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Answer 1 · 2015-03-29T00:14:49

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Solve each function for #x#

#x=21-3y#

#x=y^2-7 #

Now find out where the two curves intersect.

#y^2-7=21-3y#

#y^2+3y-28=0 #

#(y+7)(y-4)=0 #

#y=-7 # and #y=4 #

The integral for the area is

#int_-7^4 21-3y-(y^2-7)dy #

#int_-7^4 21-3y-y^2+7dy #

#int_-7^4 28-3y-y^2dy #

Integrating we get

#28y-3/2y^2-1/3y^3 #

Now evaluate

#[28(4)-3/2(4)^2-1/3(4)^3]#
#-[28(-7)-3/2(-7)^2-2/3(-7)^3]#

#[112-24-64/3]-[-196-147/2+343/3]#

#[200/3]-[-931/6] =200/3+931/6=1331/6~=221.83#