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Find the volume of the solid below the graph of the function $z = 81-x^2-3y^2$ above the region D in the xy-plane where D is the region between the parabola $y^2 = 2x+4$ and the line $y = x-1$.

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First of all find the area $D$ by intersect to functions lying on the $xy$ plane: $$y^2=2x+4\\\ y=x-1$$ You get $$y_1=1-\sqrt{7},~~y_2=1+\sqrt{7},~~x_1=2-\sqrt{7},~~x_2=2+\sqrt{7}$$ so you should think about the following triple integrals: $$\int_{y_1}^{y_2}\int_{\frac{y^2}2-2}^{y+1}\int_0^{81-x^2-3y^2}dzdxdy$$

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  • $\begingroup$ hello again; shouldn't it be $y^2-2$ instead of $-2$ in the dx integration? $\endgroup$ – kaine Mar 21 '13 at 20:33
  • $\begingroup$ I meant $y^2/2-2$ sorry $\endgroup$ – kaine Mar 21 '13 at 21:05
  • $\begingroup$ Would i take the integrals with respect to 1? $\endgroup$ – Michael Rametta Mar 21 '13 at 23:33
  • $\begingroup$ @kaine: Right. That is correct. It is a terrible typo. Thanks for remarking me that. :-) $\endgroup$ – mrs Mar 22 '13 at 19:52
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(Note: as I believe this to be homework I am only suppling a strong hint. Please let me know if you need more or if it doesn't work) Solve first for the intersections of the two functions only involving x and y. Say that they intersect when $y = a$ and $y = b$. You then need to solve for: $$\int_{a}^{b}\int_{y^2/2-2}^{y+1}z~dxdy$$

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