Question

1. Consider the surface of revolution that is given by the equation Z-R= -(x2 + y2)/R where [x],[y] < R/V2 . (a) Find the volume enclosed between the surface and the x-y plane. (b) Find the normal vector în and an equation for the tangent plane to the surface at i = ? (î+ ſ + Â). (Hint: Choose appropriate coordinate systems in each part).

Answer 1

1) Given equation Of Plane ZERO-Cory where 10), slyl < R15 Ř a g < R/ a) Volume enclosed between xy - Plane so OSIER - (2442) take za o ( defined on xy-piane, • (x2442) - R ya - R² - 12 y = SR - 22 O< y < JR? x² and N2 = R² x = IR Volume Formula - Sf S dv b 9(1) fa (ay) - I dzdy dx - agila) filmypaty?) R² 12 R- R => I dz dydx RE R o o

Q R2-42 R- 42 dyda [ ISR - (2442) 7 dy dt - R VR²_x2 Sf (R² _ x2 - y2) dy dx Frien PVR (PA) de Vio R2_x da x2 da d2 NAT da XVR²_2 + S Xal 7

ATM TRUS Par 662-233) 2 + sno Dogsp (00 } - O+ Rs ] Rog se ) 3-50785 26-'s (P) [-handry 2 - 30-TE + 30 - 3 + qups N + 87R 4 16 २ 703 - 40

(Ans) > IT (R - R²) is volume (Depends on R) b) Le FCON,Z) + z-R+ x2+y? = 2-R + x + 2 Norrmal vecto Normal vectore ń - (pag . 1) *ga2 +291 + Now to find tangent Plane to the surface de cê tŷ të) 12 1 R12 TE TE & ON 4

Ans ) R (x+y+z) = 0 be the Tangent Plane surface M = Rrêt ? t to the I (Ans)