Test 4: Sample Solutions

MAT237 9 April 2020

Test 4: Sample Solutions

1. Let

S ={(x, y) : 0 ≤y≤cos(πx) + 1,0≤x≤2} ∪ {(x, y) : (x−2)²+ (y−1)² ≤1, x≥2}

Write the area of S as iterated integrals (you do not need to evaluate).

2. Determine the volume of the solid S ⊂ R³ bounded by z = x²y²(x² +y²) and the xy-plane, and with 1 ≤xy ≤4,1≤x²−y² ≤2, x, y >0 .

3. Suppose S ⊆R³ is the intersetion of B(0,2) and the cylinder{(x, y, z) :y²+z² ≤1}, and that the density of S is given by ρ(x, y, z) = 1

x²+ 1. Set up an iterated integral which gives the mass of S (you do not need to evaluate it).

4. Define F :R³ →R by F(x) =

Z 1

0

Z 1

0

sin(y² +z²) +x²y²+x³z dydz Prove thatF is C² and determine F⁰⁰(x).

5. Let F(x, y) =

−y

x²+y², x x²+y²

. Evaluate R

CF ·dx, where C is the unit circle,

∂B(0,1), oriented coutner-clockwise.

6. Let C be the path from (0,0) to (1,0) along the curve y = x−x², and let F(x, y) = (x²+y²,2xy+e^y). Determine R

CF·dx.

7. Let S be the surface given by the graph of g(x, y) = 1−x²−y² with (x, y)∈B(0,1).

LetF(x, y, z) = (x², y, z). Set up, but do not evaluate, an iterated integral to compute RR

SF·ndA, wheren is oriented up (in the positive z direction).

8. Suppose that f :R³ →R is C¹ and F:R³ →R³ is C². Show that div(fcurlF) = ∇f·curlF

9. Suppose f :R³ →R, andF:R³ →R³. Determine whether the following compositions give a functionR³ →R(scalar-valued, write “S”), a vector fieldR³ →R³ (write “V”), or are not defined (write “DNE”). No justification is required.

(a) curl (∇(divF)) (b) curl (∇(divf)) (c) ∇(div (F) +f) (d) div (curl (∇f))

1

and also that S ⊂ R

× R

(0, v) : v≤0