Claire H.

asked • 08/16/18

Double integral - calculating area

Calculate the area of the region bounded by the two curves, y = 4 - x2 and y = x + 2.
The answer is meant to be 9/2 but I just cannot get it!!! Maybe my bounds are incorrect.... help!

3 Answers By Expert Tutors


Doug C. answered • 08/17/18

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Mark M.

I answered this question, but my answer disappeared for some reason.  Here is a summary of my answer:
   The curves intersect when 4-x2 = x+2.  Solving this equation, we get x = -2 or 1.
   Between x=-2 and x=1, the upper boundary is y = 4-x2 and the lower boundary is y = x+2.
   Area between the curves = ∫(from -2 to 1) [(4-x2) - (x+2)]dx = ∫(from -2 to 1)[2 - x - x2]dx
             Evaluate the integral to get 9/2.  


Claire H.

Sorry but I don't understand how you can just use 4-x2 and x+2 straight off the bat. Would you not have to split them because of where they "enter" and "exit" if you understand my meaning?
What I did, but it gives me the wrong answer is:
(from 0 to 4)  ∫  (from (y-2) to (-√(4-y)) dxdy + ∫(from 2 to 4) ∫ (from (y-2) to (√4-y) 
but this keeps giving me the incorrect answer and I cannot see why


Paul M.

First of all, you should make a graph of the area in question so that you can see what you are trying to compute.
Secondly, the reason you are having trouble is that the area in the 2nd quadrant will have a negative sign (because the abscissa is negative and the ordinate is positive).
You can solve the problem the way I did or you can integrate in 2 pieces.
You need to integrate the function which is the difference between the parabola and the straight line, between -2 and 0 and take the absolute value of the result and then add to that the integral between 0 and 1.
If what I have told you still doesn't help, post another "comment" and I will try to respond further.


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