How do you find the area of the region bounded by the curves #y=|x|# and #y=x^2-2# ?

Answer 1

This is a definite integral problem

First we need to find the intersections of these 2 functions by setting them equal to each other.

#|x|=x^2-2#

This has to be split up into 2 equations.

#x=x^2-2# AND #-x=x^2-2#
FIRST #x=x^2-2# #x^2-x-2=0# Factor #(x-2)*(x+1)=0# #x-2=0# #x=2#
#x+1=0# #x=-1#
SECOND #-x=x^2-2# #x^2+x-2=0# Factor #(x+2)*(x-1)=0# #(x+2)=0# #x=-2#
#(x-1)=0# #x=1#
The interval is from #[-2,2]#
The function #|x|# is greater than the function #x^2-2# over the interval [-2,2].
#int_(-2)^(2)|x|-(x^2-2)dx=int_(-2)^(2)|x|-x^2+2dx#
#=int_(-2)^(2)|x|dx-int_(-2)^(2)x^2dx+int_(-2)^(2)2dx#
Note: |x| is an even function and symmetric about the #y#-axis so make this function easier to manage we can assume x>0 and double the area found by multiplying it by 2 and changing the boundaries from 0 to 2.
#=2*int_0^(2)xdx-int_(-2)^(2)x^2dx+int_(-2)^(2)2dx#
#=2*[x^2/2]_0^2 -1* [x^3/3]_(-2)^2+1*[2x]_(-2)^2#
#=2*[(2)^2/2-(0)^2/2] -1* [(2)^3/3-(-2)^3/3]+1*[2(2)-2(-2)]#
#=2*[2-0] -1* [8/3+8/3]+1*[4+4]#
#=2*[2] - [16/3]+[8]#
#=[4] - [16/3]+[8]#
#=[12/3] - [16/3]+[24/3]#
#= [-4/3]+[24/3]#
#= [20/3]#
#= 6 2/3 or 6.6667#
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Answer 2

To find the area of the region bounded by the curves (y=|x|) and (y=x^2-2), you first need to determine the points of intersection between the two curves. Set the equations equal to each other and solve for (x). Then, integrate the absolute difference between the curves from the leftmost point of intersection to the rightmost point of intersection with respect to (x).

The points of intersection are found by setting (|x| = x^2 - 2). Solve this equation for (x) to find the (x)-coordinates of the points of intersection. Then integrate the absolute difference between the curves from the leftmost to the rightmost point of intersection to find the area of the region bounded by the curves.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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