Question

Find the volume of an object enclosed by cylinders z = y² +2, z = 4−y² and planes x = −1 and x = 2.

Answer 1

Let's find the integration boundaries:

$-1\le x\le2$ ;

to find the bounds for y equate $y^2 +2$ and $4-y^2$ :

$y^2+2=4-y^2$

$2y^2=2$

$y=\pm1$

$-1\le y\le1$

$y^2+2\le z\le 4-y^2$

$\displaystyle V=\int_{-1}^2dx\int_{-1}^1 dy\int_{y^2+2}^{4-y^2}dz=$$\displaystyle \int_{-1}^2dx\int_{-1}^1 (4-y^2-(y^2+2))dy=$$\displaystyle \int_{-1}^2dx\int_{-1}^1 (2-2y^2)dy=$$\displaystyle \int_{-1}^2dx (2y-\frac23y^3)|_{-1}^1=\displaystyle \frac83\int_{-1}^2dx =$$\displaystyle \frac83(2-(-1))=8$

Answer: $V=8$.