$$ S = \int_{-8}^{8} \int_{-\frac{21}{2}}^{21} \left(1 + \frac{1}{8} x^2 + \frac{64}{3969} y^2\right)^{\frac{1}{2}} dy \, dx $$
I'm trying to finish a high school assignment, and because of the question I chose for myself, where I'm calculating the area of a surface above some region $R$ in the $xy$-plane, it turned into this double integral that I have tried to solve.
My problem is with calculating the inner integral, where I cannot use the reverse chain rule for single-variable integrals because there are two variables, and I'm unsure what other methods I can use. Any help would be greatly appreciated, thanks!
$$S = \int_{-8}^{8} \int_{-\frac{21}{2}}^{21} \left(1 + \frac{1}{8} x^2 + \frac{64}{3969} y^2\right)^{\frac{1}{2}} dy \, dx= \int_{-8}^{8} \int_{-\frac{21}{2}}^{21} \left(1 + \left(\frac{1}{2\sqrt{2}} x\right)^2 + \left(\frac{8}{63} y\right)^2\right)^{\frac{1}{2}} dy \, dx $$ $$u=\frac{1}{2\sqrt2}x, v=\frac{8}{63}y$$ $$S={2\sqrt2}\cdot\frac{63}{8}\cdot\int_{-2\sqrt2}^{2\sqrt{2}}\int_{-\frac43}^\frac83\sqrt{1+u^2+v^2} dv \, du$$