Calculus 3

Math 3C Quiz 5 Sections 16.1 to 16.3

Name________________ Show all work!

1) Determine the definite integral that results from the line integral ∫ 𝑓 𝑑𝑠! , where the curve C is parameterized by r(𝑡) = ⟨sin(𝑡) , ln (𝑡)⟩, for 1 ≤ 𝑡 ≤ 𝜋, and 𝑓(𝑥,𝑦) = 𝑥" + 𝑥𝑦. Do not evaluate the integral.

2) Determine the definite integral that results from the line integral ∫ F ∙ 𝑑r! , where the curve C is parameterized by r(𝑡) = ⟨cos(𝑡) ,𝑡#,25𝑡⟩, for 0 ≤ 𝑡 ≤ 19, and the field is 𝐅(𝑥,𝑦,𝑧) = ⟨𝑥,𝑥𝑦,𝑥𝑦𝑧⟩. Do not evaluate the integral.

3)

4) Evaluate the line integral ∫ F ∙ 𝑑r! , where the curve C is parameterized by r(𝑡) = ⟨ln(t) ,𝑒 − 𝑡⟩, for 1 ≤ 𝑡 ≤ 𝑒, and the field is 𝐅(𝑥,𝑦) = ∇𝑓, where 𝑓(𝑥,𝑦) = tan(𝑥𝑦) + 𝑥 + 2𝑦.