How Do You Find The Area Between F(x)=x^2+2x+1,g(x)=3x+3?

How do you find the area between #f(x)=x^2+2x+1,g(x)=3x+3#? Calculus Using Integrals to Find Areas and Volumes Calculating Areas using Integrals

2 Answers

Anjali G Nov 16, 2016

4.5 square units

Explanation:

The points of intersection of the two functions are at: #(-1,0)# and #(2,9)#.

So, we have to find the area under the curve of #g(x)#, then subtract the area under the curve of #f(x)# because #g(x)# is above #f(x)# in the interval #[-1,2]#. #int_-1^2g(x)dx-int_-1^2f(x)dx# #int_-1^2(3x+3)dx-int_-1^2(x^2+2x+1)dx#

Simplify the are under the curve of the linear equation using geometry, and the area under the curve of the parabola using a fundamental theorem of calculus.

Let #F(x)# be the antiderivative of #f(x)# #1/2(3)(9)-[F(2)-F(-1)]# #27/2-[F(2)-F(-1)]#

To simplify this, we need to know #F(x)#. Integrate #f(x)#: #F(x)=intx^2+2x+1# #F(x)=1/3x^3+x^2+x+c# Solve #F(2)# and #F(-1)#: #F(2)=1/3(2)^3+2^2+2+c=26/3+c# #F(-1)=1/3(-1)^3+(-1)^2-1+c=-1/3+c#

Now continue to simplify the expression above: #27/2-[F(2)-F(-1)]# #13.5-[(26/3cancel(+c))-(-1/3cancel(+c))]# #=13.5-9# #=4.5# square units

Answer link Noah G Nov 16, 2016

Start by finding the intersection points of the two curves.

#{(y = x^2 + 2x + 1), (y = 3x + 3):}#

#3x + 3 = x^2 + 2x + 1#

#0 = x^2 - x - 2#

#0 = (x - 2)(x + 1)#

#x = 2 and -1#

#y = 3x + 3#

#y = 9 and 0#

Hence, the points of intersection are #{2, 9}# and #{-1, 0}#

We now do the graph of each function (on the same grid). We find that #g(x)# is above #f(x)# in the area that they share, so we find the area under #g(x)#, and will subtract the area of #f(x)# from that.

#int_(-1)^2 (3x + 3)dx#

#=>3/2x^2 + 3x|_(-1)^2#

#=>3/2(2)^2 + 3(2) - (3/2(-1)^2 + 3(-1))#

#=>3/2(4) + 6 - 3/2 + 3#

#=>6 + 6 - 3/2 + 3#

#=>27/2#

Now for #f(x)#:

#int_(-1)^2 (x^2 + 2x + 1)#

#=>1/3x^3 + x^2 + x|_(-1)^2#

#=>1/3(2)^3 + 2^2 + 2 - (1/3(-1)^3 + (-1)^2 - 1)#

#=> 8/3 + 4 + 2 + 1/3 - 1 + 1#

#=>9#

Now subtract the two areas.

#=>27/2 - 9 = 9/2 = 4.5#

Hence, the area between the graphs of #f(x) = x^2 + 2x + 1# and #g(x) = 3x + 3# is #4.5" units"^2#.

Hopefully this helps!

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