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 $y = x-1$.


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: $$\int_{y_1}^{y_2}\int_{\frac{y^2}2-2}^{y+1}\int_0^{81-x^2-3y^2}dzdxdy$$

enter image description here

  • $\begingroup$ 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

(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: $$\int_{a}^{b}\int_{y^2/2-2}^{y+1}z~dxdy$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.