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Calc 3 13.6

 

<== Calculus 3
Number Question Answer
See the work
1. Consider x = h(y,z) as a parametrized surface in the natural way. Write the equation of the tangent plane to the surface at the point (2, 2, 2) given that rac{\partial h}{\partial y}(2,2) = -4 and rac{\partial h}{\partial z}(2,2) = 5.
.
x-2+4(y-2)-5(z-2) = 0 The work
2. For the surface with parametric equations r(s,t) = &lang st, s + t, s - t &rang, find the equation of the tangent plane at (2,3,1).
.

Find the surface area under the restriction s^2 + t^2 \leq 1

Equation of tangent plane:
-2(x-2)+3(y-3)-(z-1)=0

surface area: 7.013
The work
3. Find the surface area of that part of the plane 2x + 10y + z = 10 that lies inside the elliptic cylinder rac{x^2}{16} + rac{y^2}{25} =1
Surface Area =
643.83 The work
4. Write down the iterated integral which expresses the surface area of z = y^{2}cos^{5}x over the triangle with vertices (-1,1), (1,1), (0,2):

\int_a^b\int_{f(y)}^{g(y)} \sqrt{h(x,y)}\,dx dy

a =
b =
f(y) =
g(y) =
h(x,y) =
a = 1
b = 2
f(y) = y-2
g(y) = 2-y
h(x,y) =
The work
5. The vector equation r(u,v) = ucos(v)i + usin(v)j + vk , 0 &le v &le 8&pi, 0 &le u &le 1, describes a helicoid (spiral ramp). What is the surface area?

28.85 The work
6. Find the surface area of the part of the sphere x^{2} + y^{2} + z^{2} = 4 that lies above the cone z = \sqrt{x^{2} + y^{2}}

7.36 The work
7. Find the area cut out of the cylinder x^2 + z^2 = 1 by the cylinder x^2 + y^2 =  1.
8 The work
8. If a parametric surface given by \mathbf{r_{1}}(u, v) = f(u, v)\mathbf{i} + g(u, v)\mathbf{j} + 
  h(u, v)\mathbf{k} and -4 &le u &le 4, -5 &le v &le 5, has surface area equal to 5, what is the surface area of the parametric surface given by \mathbf{r_{2}}(u, v) = 5\mathbf{r_{1}}(u, v) with -4 &le u &le 4, -5 &le v &le 5?

125 The work
9. question Answer The work
10. question Answer The work