Look at this picture. Green represents the projection of the surface on the YOZ plane, and red also represents the projection of the surface on the YOZ plane.
Then, the two surfaces intersect to form a closed curve, and the projection on the YOZ plane is the magenta line segment.
It can be seen that finding this volume is actually using double integral.
Then, the double integral is actually an expression composed of three parts, ① is the symbol of the double integral, ② is the integrand, ③ is the differential of the integral area, and that will do.
Now the integral area must be the closed area of the closed curve surrounded by the intersection of two surfaces on the XOY plane, so we can say that the two surface equations are simultaneous equations and get x2+y2=2. It can be seen that it is a circle.
Therefore, the integral area d of double integral can be expressed as: x∈[-√2, √2], y∈[-(√2-x2), (√2-x2)].
Now, there are only integrand functions.
Still looking at the figure, combined with the analysis of the curved top cylinder of double integral, we should know that the integrand function is actually the height.
So for two surfaces, I don't know which surface is above and which surface is below. How to find the height?
It goes up and down whatever it asks, and I think one is up and down. If it is a negative number, the absolute value is over.
Then, the integrand function is directly expressed as 6-2x2-y2-(x2+2y2), depending on the result, so according to the definition of double integral, there are
Double integer (6-3x2-3y2)dxdy
Then according to the integral area calculation results.