(a) i. Evaluate $$int_{0}^{5} int_{0}^{sqrt{25-x^{2}}} int_{0}^{6} frac{1}{sqrt{x^{2}+y^{2}}} dz dy dx$$ ii. Convert to spherical coordinates and evaluate $$int_{0}^{1} int_{0}^{sqrt{1-x^{2}}} int_{0}^{sqrt{1-x^{2}-y^{2}}} frac{1}{1+x^{2}+y^{2}+z^{2}} dz dy dx$$ (b) Find the area of the surface part of the paraboloid $z = x^2 + y^2$ that lies under the plane $z = 9$ that is bounded above $z = 9$ (c) Evaluate the integral $$int_{z=0}^{z=1} int_{y=2}^{y=4} int_{x=-1}^{x=5} (x + yz^2) dx dy dz$$ (d) Evaluate $$int_{0}^{1} int_{0}^{e^x} (10 + x^2 - y^2) dy dx$$
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The integral is given by \[ \int_{0}^{5} \int_{0}^{\sqrt{25} x^{3}} \int_{0}^{6} \frac{1}{\sqrt{x^{2}+y^{2}}} d z d y d x \] This integral is not well-defined because the integrand is not defined when \(x = y = 0\). ii. The integral is given by \[ \int_{0}^{1} Show more…
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