How to Solve a Double Integral with an Elliptical Region?

[Oster]

1. Find the area of the ellipse (2x + 5y − 3)^2 + (3x − 7y + 8)^2 < 1

I have no idea what this looks like, and hence I can't figure out the limits. Maybe I could transform it into a more familiar form using a translation and rotation? Please help.

 

[Char. Limit]

Have you learned about the Jacobian? This seems like a good place to use it. In essence, let u=2x+5y-3 and v=3x-7y+8. It should give you a nice coordinate transformation that will change your ellipse to something easily integrable.

 

[Oster]

Thanks, it turned out to be really easy!