91Ó°ÊÓ

$$f(x, y,z) = x − y − z$$ on the portion of the plane with equation $$10x − \sqrt {33y} + 36z = 30$$ with $$2 ≤ x ≤ 7$$ and $$1 ≤ z ≤ 2$$.

We have, $$10x − \sqrt {33y} + 36z = 30$$

$$\implies y=\frac{10x}{\sqrt {33}}+\frac{36z}{\sqrt {33}}-\frac{30}{\sqrt {33}}$$

Here, $$ds=\sqrt{(\frac{10x}{\sqrt {33}})^{2}+(\frac{36z}{\sqrt {33}})^{2}-(\frac{30}{\sqrt {33}})^{2}}dA$$

$$\implies ds=\sqrt{\frac{1429}{33}}dxdz$$

Now, evaluating the given integral, we get

$$\int_{s}f(x,y,z)ds=\int x-(\frac{10x}{\sqrt {33}}+\frac{36z}{\sqrt {33}}-\frac{30}{\sqrt {33}})ds$$

$$\implies \int_{s}f(x,y,z)ds=\int_{1}^{2}\int_{2}^{7} (x-\frac{10x}{\sqrt {33}}+\frac{36z}{\sqrt {33}}-\frac{30}{\sqrt {33}})\sqrt{\frac{1429}{33}}dxdz$$