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Practice Problems for Assignment - Calculus III | MATH 210, Assignments of Advanced Calculus

University of Illinois - ChicagoAdvanced Calculus

Material Type: Assignment; Class: Calculus III; Subject: Mathematics; University: University of Illinois - Chicago; Term: Spring 2012;

Typology: Assignments

2011/2012

Uploaded on 05/18/2012

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koofers-user-9uw 🇺🇸

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MATH 210
Homework due 03/14/2012
1. Show that the Jacobian of the transformation from Cartesian to spher-
ical coordinates is indeed ρ2sin φ.
2. Let Dbe the quadrilateral defined by the lines:
y=x y = 2x x +y= 3 x+y= 6
Use the change of variable x=u
v+ 1 and y=uv
v+ 1 in order to
compute the integral:
ZZD
(x+y)dA
3. Let Dbe the quadrilateral defined by the lines:
x+ 2y= 10 x+ 2y= 6 y= 3 y= 1
Use the change of variable x=u−vand y=vin order to compute
the integral:
ZZD
(x+ 3y)dA
4. Find the critical points of the function
f(x, y) = x3+y2−6xy −6x2
and classify as (local) maximum, minimum or saddle points.
5. Find the volume of the solid in the first octant bounded by the planes:
x= 0, y = 0, z = 0, x + 2y+z= 6
6. Sketch the region of integration and change the order of integration,
but DO NOT EVALUATE:
Z3
0Z3
√y
f(x, y)dxdy
1
pf2

Partial preview of the text

Download Practice Problems for Assignment - Calculus III | MATH 210 and more Assignments Advanced Calculus in PDF only on Docsity!

MATH 210

Homework due 03/14/

  1. Show that the Jacobian of the transformation from Cartesian to spher- ical coordinates is indeed ρ^2 sin φ.
  2. Let D be the quadrilateral defined by the lines:

y = x y = 2x x + y = 3 x + y = 6

Use the change of variable x =

u v + 1

and y =

uv v + 1

in order to compute the integral: ∫ ∫

D

(x + y)dA

  1. Let D be the quadrilateral defined by the lines:

x + 2y = 10 x + 2y = 6 y = 3 y = 1

Use the change of variable x = u − v and y = v in order to compute the integral: ∫ ∫

D

(x + 3y)dA

  1. Find the critical points of the function

f (x, y) = x^3 + y^2 − 6 xy − 6 x^2

and classify as (local) maximum, minimum or saddle points.

  1. Find the volume of the solid in the first octant bounded by the planes:

x = 0, y = 0, z = 0, x + 2y + z = 6

  1. Sketch the region of integration and change the order of integration, but DO NOT EVALUATE:

∫ (^3)

0

√y^ f^ (x, y)dxdy

  1. Determine whether the vector field F(x, y) =< 4 x^3 exy, x^4 yexy^ > is conservative or not.
  2. Compute the integral (^) ∫ ∫

D

x^2 y + 1

dA

where D is the triangle with vertices (0, 1), (1, 2), (3, 0).

  1. Compute the integral (^) ∫ ∫

D

x^4 dA

where D is the unit disk {(x, y) : x^2 + y^2 ≤ 1 }

  1. Apply the change of variable x = u + 2v y = 3u − v in order to com- pute the integral of the function f (x, y) = 3x^2 y over the quadrilateral with vertices (0, 0), (1, 3), (2, −1) and (3, 2).
  2. Compute the integral of f (x, y, z) = (x^2 + y^2 + z^2 )^2 over the ball centered at (0, 0 , 0) with radius 2.
  3. Find the critical points of the function f (x, y) = x^4 + y^4 + 4xy − 1 and classify them as maximum, minimum or saddle points.
  4. Let f (x, y, z) = 1 + x^3 + y^2 − z^3. Suppose you were using the method of Lagrange multipliers to find the maximum value of the function f on the ellipsoid x^2 + 3y^2 + 2z^2 = 3. a) Write down the system of 4 algebraic equations in 4 unknowns tha you would need to solve. Do not try to solve these equations. b) State how you would find the maximum value, given the list of solutions to the equations in part (a).
  5. Change the order of integration to compute the iterated integral:

∫ (^3)

0

x/ 3

ey

3 dydx

  1. Find the minimum and maximum of the function f (x, y, z) = x^2 − y^2 + 2z^2 when x^2 + y^2 + z^2 = 1.
  2. Determine the local extrema of the function f (x, y, z) = 2x^3 − 3 y^2 + xy + 2.