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Calculus III - Fall 2011 Test 3, Exams of Advanced Calculus

Symbiosis InternationalAdvanced Calculus

The calculus iii test held in fall 2011. The test consists of 10 questions, each worth 10 points. The questions involve various integration techniques, including finding limits, volumes, and mass of laminae. Students are required to sketch regions of integration and change the order of integration. Some questions also involve transforming coordinates.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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CALCULUS III 1
MA 227, CALCULUS III
Fall, 2011
Name (Print last name first): ..........................................
Student Signature: ...................................................
TEST 3
10 questions, 10 points each. SHOW ALL YOUR WORK!
Question 1
Find R RDx dxdy, where Dis bounded by y=x3and y=x2.
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Download Calculus III - Fall 2011 Test 3 and more Exams Advanced Calculus in PDF only on Docsity!

MA 227, CALCULUS III

Fall, 2011

Name (Print last name first):..........................................

Student Signature:...................................................

TEST 3

10 questions, 10 points each. SHOW ALL YOUR WORK!

Question 1

Find ∫ ∫ D x dxdy, where D is bounded by y = x^3 and y = x^2.

Question 2

Find the volume under the surface z = y and above the triangle in the xy plane with

vertices (0, 0), (2, 0), (1, 1).

Question 4

Use polar coordinates to find the volume under the plane z = 2x − y + 6 and above

the half-disk x^2 + y^2 ≤ 4 , y ≥ 0 in xy plane.

Question 5

Find the mass of the lamina that occupies the region:

D = {(x, y)| x^2 + y^2 ≤ 1 , x ≤ 0 }

and has the density function given by ρ(x, y) = x^2 + y^2.

Question 7 Express the integral ∫ ∫ ∫ E f (x, y, z)dV as an iterated integral, where E is the solid above the region D = {(x, y) : y^2 ≤ x ≤ y} in xy plane and below the plane z = x + y.

Question 8 Find ∫ ∫ D(x − 2 y) dxdy, where D is bounded by x + y = 0, x + y = 4, x − 2 y = 1, x − 2 y = 2. Use change of variables u = x + y, v = x − 2 y.

Question 10 (a) Change (2, π/ 4 , π/3) from spherical to rectangular coordinates.

(b) Change (1, −√ 3 , 0) from rectangular to spherical coordinates.