Double integral - calculating area

Question

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!

Answer 1

Hi Claire,

 

Take a look at this graph to get the idea of what is happening. This problem is an example of the "area between two curves". To find the height of a "typical rectangle" use "top" - "bottom".

 

https://www.desmos.com/calculator/mmynniltq1

 

Hopefully the above graph will give you some ideas on how to use Desmos to get a feel for what is happening for problems like this.

Answer 2

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Answer 3

I think that you mean the area bound ONLY by the parabola an the straight line.

Part of this area will have a negative sign; therefore you have to find the area in pieces.

The area enclosed by the parabola and the x-axis is twice the area bounded by the parabola, the y-axis and the x-axis to the right of the origin, i.e. twice the integral from 0 to 2 of 4 - x²; that area is 32/3

The triangular area bounded by the straight line and x = 1 is 9/2.

the area bounded by the parabola, x = 1 and the x-axis is the integral of 4 - x² between 1 and 2; this area is 5/3.

 

The area required then is (32/3) - (9/2) - (5/3) = 9/2.