Finding the Area Between Two Curves Using Jacobians

[Saitama]

Homework Statement

Consider curves $C_1: (y-x)=(x+y-\sqrt{2})^2$ and $C_2: (x+y-\sqrt{2})=(y-x)^2$, then the area between $C_1$ and $C_2$ is
A)1/2
B)1/3
C)1/4
D)None

Homework Equations

The Attempt at a Solution

Finding out the points of intersection would be a lot difficult here. And even if I find them, integration would be dirty. This is a question from my test paper and I suspect that it has an easy solution but I am unable to figure that out.

Any help is appreciated. Thanks!

 

[Dick]

Homework Statement

Consider curves $C_1: (y-x)=(x+y-\sqrt{2})^2$ and $C_2: (x+y-\sqrt{2})=(y-x)^2$, then the area between $C_1$ and $C_2$ is
A)1/2
B)1/3
C)1/4
D)None

Homework Equations

The Attempt at a Solution

Finding out the points of intersection would be a lot difficult here. And even if I find them, integration would be dirty. This is a question from my test paper and I suspect that it has an easy solution but I am unable to figure that out.

Any help is appreciated. Thanks!

Did you think about trying a change of variables?

 

[Saitama]

Did you think about trying a change of variables?

Nope. How would I do that here? Substitute $y-x$ with $t$?

 

[Dick]

Nope. How would I do that here? Substitute $y-x$ with $t$?

Sure. Call x+y+sqrt(2)=s, x-y=t. Find the area is s,t coordinates. Don't forget the Jacobian factor.

 

[Saitama]

...Jacobian factor.

Sorry, never heard of that before.

Hmm...using the substitution, the question is similar to finding area between $y=x^2$ and $y^2=x$. The area between them is 1/3. This is the answer given in the answer key. Thank you Dick!

 

[Dick]

Sorry, never heard of that before.

Hmm...using the substitution, the question is similar to finding area between $y=x^2$ and $y^2=x$. The area between them is 1/3. This is the answer given in the answer key. Thank you Dick!

See http://www.math.oregonstate.edu/hom...ulusQuestStudyGuides/vcalc/change/change.html about Jacobians. I don't think 1/3 is actually correct.

 

[Saitama]

See http://www.math.oregonstate.edu/hom...ulusQuestStudyGuides/vcalc/change/change.html about Jacobians. I don't think 1/3 is actually correct.

I don't think I can understand that and multiple integrals aren't in my syllabus. So the answer in key is wrong?