Description
Do problems 5.245, 5.258, and 5.273. Those require double and triple integration
Explanation & Answer
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Double Integration Excercise
5.258
Surface S is defined by the parametric equations 𝑥 = 𝑢 + 𝑣, 𝑦 = 𝑢2 , 𝑧 = 𝑢(𝑣 + 1)
for 0 ≤ 𝑢 ≤ 2, 0 ≤ 𝑣 ≤ 5 . Find ∬𝑆 𝑓⃗. 𝑑𝐴⃗ for𝑓⃗ = 𝑧 2 𝑖̂ + 𝑧 2 𝑗̂ + (𝑥𝑦)𝑘̂.
a) Begin by using the equation for surface S to rewrite 𝑓⃗ as a function of u and v
b) Rewrite surface S as a vector-valued function 𝑟⃗(𝑢, 𝑣).
c) Use equation 5.10.2 to find 𝑑𝐴⃗ for the surface S
d) Write a double integral for ∬𝑆 𝑓⃗. 𝑑𝐴⃗
e) Evaluate the double integral
Solution
a) 𝑓⃗ can be rewritten as
𝑓⃗ = 𝑧 2 𝑖̂ + 𝑧 2 𝑗̂ + (𝑥𝑦)𝑘̂
̂
= 𝑢2 (𝑣 + 1)2 𝒊̂ + 𝑢2 (𝑣 + 1)2 𝒋̂ + 𝑢2 (𝑢 + 𝑣)𝒌
b) The surface S...
Review
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