# Finding the volume of $z = 81-x^2-3y^2$

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: $$\int_{y_1}^{y_2}\int_{\frac{y^2}2-2}^{y+1}\int_0^{81-x^2-3y^2}dzdxdy$$
• hello again; shouldn't it be $y^2-2$ instead of $-2$ in the dx integration? – kaine Mar 21 '13 at 20:33
• I meant $y^2/2-2$ sorry – kaine Mar 21 '13 at 21:05
(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: $$\int_{a}^{b}\int_{y^2/2-2}^{y+1}z~dxdy$$