Question

1.

A solid in the first octant, bounded by the coordinate planes, the plane (x= 40) and the curve (z=1-y² ) , Find the volume of the solid by using : a- Double integration technique ( Use order dy dx) b-Triple integration technique ( Use order dz dy dx)

2.

Use triple integration in Cartesian coordinates to find the volume of the solid that lies below the surface = 16 − ?² − ?² , above the plane z = 40/ 100 , and bounded by the curve ? = √? and the lines ? = ? − 2 and ? = −?.

3.

Find all the local maxima, local minima, and saddle points of the function ?(?, ?) = 40?² − 2?³ + 3?² + 6?y

4.

Let ? = ? − sin(??) ?ℎ??? ? = 40? ? = ln(?) ? = e^(t-1)Find ??/?? by :- a- Using Chain Rule principles
b- Expressing (?) in term of (?) then differentiating directly

5.

a- Use implicit differentiation to find (d²y)/(dx² ) of the following curve at the point (π, 2π).y² = x²+ sin ?y

b- Show that ?⁵?⧸ ??²??³ is zero in two differentiation steps only. ?(?, ?) = ??^ ?²/⁴⁰
محمد مهدي جميل ?, [⁧يونيو ⁨30⁩، ⁨2020⁩ في ⁨18:22⁩⁩]
Show that ?⁵?/ ??²??³ is zero in three differentiation steps only. ?(?, ?) = ?² + ?(sin ? – ?⁴ ).

Answer 1