Math 210 Summer 2023 Exam #3
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Math 210
Summer 2023
Exam #3
Leave all answers in radical form. Good Luck!1)Sketch the region R whose area is given by the iterated integral and calculate its area (12 pts)
2) Suppose region R is defined in the first quadrant by {(x, y)| 4 ≤ x2 + y2 ≤ 9}. Sketch R and use f(x, y) = √9 − x2 − y2 in polar to evaluate the following. (13 pts)
3) Evaluate the following (12 pts)
4) Evaluate the following line integral on C, the line segment from (0,0,0) to (1,2,1) leave your answer in radical form (15 pts)
5) Show that the vector field (x, y) = 2xy + (x2 − y) is conservative and find a potential function f(x,y). (12 pts)
6) Use Green’s Theorem to evaluate the line integral
Where C is the path from (0,0) to (1,1) along the graph of y = x 3 and from (1,1) to (0,0) along the graph of y = x. (16 points)
7) Find the Divergence of (x, y, z) = ex + yz − yz2 and then evaluate it at (ln(3),2,4) (10 pts)
8) Find the curl of(x, y, Z) = 2xy + (x3 + Z 2 ) + 2yZ (10 pts)
Bonus: Consider Vector Field (x, y, Z) = − x − y +
Evaluate the line integral ∙ d
on path C : (t) = cost + sint + t from (1,0,0) to (−1,0,3π)
(10 pts)
2023-08-08