find the area of the region that is

Answered question

2022-05-05

find the area of the region that is bounded by the graphs of x = y^2 and x = 2 − y

Answer & Explanation

user_27qwe

user_27qwe

Skilled2022-07-02Added 375 answers

Based on the sketch. we are looking for a double integral solution to calculate the area bounded by the curves:

x=y2
y=x+2=>x=y2

The points of intersection are the solution of the equation:

x =  ( x + 2 )2

 x =  ( x2 + 4 x + 4 )  

 x2 + 5 x + 4 = 0

(x+1)(x+4) = 0

x=-1, -4

The corresponding y-coordinates are:

x =  1  y = 1  

x =  4  y =  2

Giving the coordinates (−1,1) and (−4,−2)

If in the above diagram we look at an infinitesimally thin horizontal strip (in black) then the limits for x and y are:

x varies from y−2 to y2
y varies from −2 to 1

And so we can represent the bounded are by the following double integral:

A=RdA
   =-21y-2-y2 dx dy

We can calculate the inner integral:

y2y2dx=[x]y-2 

=(y2)(y2)
=y2y+2
=(y2+y2)

And so:

= -21-( y2+y2 ) dy  

=92

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