Question

Use a graphing device to draw the solid enclosed by the paraboloids \({\rm{z = }}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}\) and \({\rm{z = 5 - }}{{\rm{x}}^{\rm{2}}}{\rm{ - }}{{\rm{y}}^{\rm{2}}}.\)

Short answer

They have circular paraboloids \({\rm{z = }}{{\rm{r}}^{\rm{2}}}\) opening upwards and circular paraboloids \({\rm{z = 5 - }}{{\rm{r}}^{\rm{2}}}\) opening downwards.

Step-by-step solution

the solid enclosed by the paraboloids,

Use polar coordinates,

Can write the two equations in polar form.

The graphs will intersect when \({\rm{5 - }}{{\rm{r}}^{\rm{2}}}{\rm{ = }}{{\rm{r}}^{\rm{2}}},\)which gives us \({\rm{r = }}\sqrt {{\rm{2}}{\rm{.5}}} \approx {\rm{1}}{\rm{.58}}\)

Therefore, they have circular paraboloids \({\rm{z = }}{{\rm{r}}^{\rm{2}}}\) opening upwards and circular paraboloids \({\rm{z = 5 - }}{{\rm{r}}^{\rm{2}}}\) opening downwards.