Download Line Integrals: Practice Problems and more Exams Calculus in PDF only on Docsity!
Line Integrals: Practice Problems
EXPECTED SKILLS:
- Understand how to evaluate a line integral to calculate the mass of a thin wire with density function f (x, y, z) or the work done by a vector field F(x, y, z) in pushing an object along a curve.
- Be able to evaluate a given line integral over a curve C by first parameterizing C.
- Given a conservative vector field, F, be able to find a potential function f such that F = ∇f.
- Be able to apply the Fundamental Theorem of Line Integrals, when appropriate, to evaluate a given line integral.
- Know how to evaluate Green’s Theorem, when appropriate, to evaluate a given line integral.
PRACTICE PROBLEMS:
- Evaluate the following line integrals.
(a)
C
(xy + z^3 ) ds, where C is the part of the helix r(t) = 〈cos t, sin t, t〉 from t = 0 to t = π π^4
(b)
C
( (^) x 1 + y^2
ds where C is given parametrically by x = 1+2t, y = t, for 0 ≤ t ≤ 1 √ 5
(π 4
- Find the mass of a thin wire in the form of y =
9 − x^2 (0 ≤ x ≤ 3) if the density function is f (x, y) = x√y. 6
- Evaluate
C
F· dr, where F(x, y) = 〈x^2 , xy〉 and C : r(t) = 〈2 cos t, 2 sin t〉 for 0 ≤ t ≤ π
0
- For each of the following, compute the work done by the vector field F on the particle that moves along C.
(a) F(x, y) = (xy)i + x^2 j where C is the portion of x = y^2 from (0, 0) to (1, 1) 3 5 (b) F(x, y, z) = (x + y)i + xyj − z^2 k where C consists of the line segment from (0, 0 , 0) to (1, 3 , 1) followed by the line segment from (1, 3 , 1) to (2, − 1 , 4) −^372
- Evaluate the following line integrals.
(a)
C
(x + 2y) dx + (x − y) dy where C : x = 2 cos t, y = 4 sin t, 0 ≤ t ≤ π 4
−^9 2
− π
(b)
C
(y − x) dx + (xy) dy where C is the line segment from (3, 4) to (2, 1)
−^392
(c)
C
y dx − x dy where C is as shown below
- For each of the following, determine whether the given vector field in the plane is a conservative vector field. If so, find a potential function.
(a) F(x, y) = 〈x, y〉 Yes; f (x, y) =^12 (x^2 + y^2 ) + C
(b) F(x, y) = 〈 3 y^2 , 6 xy〉 Yes; f (x, y) = 3xy^2 + C
(a)
C
2 xy dx + y^2 dy where C is the closed curve formed by y = x 2 and y =
x
−^6415
(b)
C
xy dx + (x + y) dy where C is the triangle with vertices (0, 0), (2, 0), and (0, 1) 1 3
(c)
C
(e^3 x + 2y) dx + (x^3 + sin y) dy where C is the rectangle with vertices (2, 1), (6, 1), (6, 4) and (2, 4). 600
(d)
C
ln (1 + y) dx − (^) 1 +xy y dy where C is the triangle with vertices (0, 0), (2, 0), and (0, 4) − 4
