MHB Lazi's question at Yahoo Answers regarding the area bounded by two functions

AI Thread Summary
The discussion focuses on calculating the area between the curves defined by the equations x=y^2 and x=4-y^2. The area is determined by shifting the functions and using symmetry to simplify the integration process. The final calculated area is confirmed to be (16√2)/3. A detailed solution is provided, illustrating the integration steps involved. This method effectively demonstrates how to find the area between the specified curves.
MarkFL
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Here is the question:

Find the area between those curves: x=y^2, and x=4-y^2?

Find the area between those curves: x=y^2, and x=4-y^2

the answer should be (16 sqrt2 ) /3

Here is a link to the question:

Find the area between those curves: x=y^2, and x=4-y^2? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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Hello Lazi,

I would orient the coordinate axes such that the $x$-axis is vertical, and the $y$-axis is horizontal. To further simplify matters, I would vertical shift both functions down two units, then use symmetry so that we wish to find 4 times the following shaded area:

2hckkl2.jpg


Hence the desired area $A$ is:

$$A=4\int_0^{\sqrt{2}}2-y^2\,dy=\frac{4}{3}\left[6y-y^3 \right]_0^{\sqrt{2}}=\frac{4}{3}\left(6\sqrt{2}-2\sqrt{2} \right)=\frac{16\sqrt{2}}{3}$$
 
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