Find the volume of the solid bounded by the elliptic

Find the volume of the solid bounded by the elliptic cylinder: $$ 4y^2+z^2=4, $$ the coordinate planes, and the plane: $$ x+y+z=5. $$

Step 1 · Identify the base region

In the first octant, the quarter-ellipse is $y^2+\dfrac{z^2}4\leqslant1,\ y,z\geqslant0$, i.e. for $0\leqslant y\leqslant1$, $0\leqslant z\leqslant2\sqrt{1-y^2}$.

Step 2 · Set up the volume integral

$$ V = \int_0^1\!\int_0^{2\sqrt{1-y^2}}(5-y-z)\,dz\,dy. $$

Step 3 · Evaluate the inner integral

$$ \int_0^{2\sqrt{1-y^2}}(5-y-z)\,dz = 2(5-y)\sqrt{1-y^2}-2(1-y^2). $$

Step 4 · Evaluate the outer integral

Splitting $\displaystyle\int_0^1(5-y)\sqrt{1-y^2}\,dy=5\int_0^1\sqrt{1-y^2}\,dy-\int_0^1y\sqrt{1-y^2}\,dy=5\cdot\frac\pi4-\frac13=\frac{5\pi}4-\frac13$ (the first piece is a quarter-circle area, the second via $u=1-y^2$), and $\displaystyle\int_0^1(1-y^2)\,dy=\frac23$:

$$ V = 2\left(\frac{5\pi}4-\frac13\right)-2\cdot\frac23 = \frac{5\pi}2-\frac23-\frac43 = \frac{5\pi}2-2. $$

Additional Notes

• This derivation matches the source exactly.
• Verified symbolically: direct evaluation of the nested integral gives exactly $\dfrac{5\pi}2-2\approx5.854$.