Problems from Areas delimited by two functions

Calculate the area enclosed between the following two functions:

f(x)=x2+4x and g(x)=x4.

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Development:

Let's first look for the points of intersection:

x2+4x=x4 x2+3x+4=0 x=3±94(1)(4)2(1)=3±252=3±52 x=1 and x=4

If we draw the functions we see that f(x) is above g(x), so the order has to be f(x)g(x):

14(x2+4xx+4) dx=14(x2+3x+4) dx=

=[x33+3x22+4x]14=433+3422+44((1)33+3(1)22+4(1))=

=643+24+16(+13+324)=1256 u2

Solution:

The area is 1256 u2

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Calculate the area enclosed by the two parables f(x)=x2+2x1 and g(x)=x2+2x.

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Development:

Let's first look for the points of intersection:

x2+2x1=x2+2x 2x21=0 x2=12 x=1±2

If we draw the parables we see that g(x) is above f(x), so the order has to be g(x)f(x):

1212(x2+2xx22x+1) dx=1212(2x2+1) dx=

=[2x33+x]1212=2(12)33+12(2(12)3312)=

=21223+12(2122312)=

=2322+12(232212)=

=132+12132+12=

=232+22=2+632=432 u2

Solution:

The area is 432 u2

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