calculas

Math 2060 Summer 1 2019 TEST 3

Printed Name:

Directions: Read each question and all directions carefully before you begin. In order to receive full credit you must

• Show legible and logical (relevant) justification, which supports your final answer.

• Use correct and complete notation.

• Simplify all answers.

• Present the answer in a mathematical equation in a proper and complete English sentence unless otherwise directed.

• Include proper units if appropriate.

• Use techniques developed in this unit. No credit will be given for the use of any formula not derived in class and approved by the instructor.

The use of any electronic devices (calculator, computer, cell phone, pda, etc), books, notes, or any other aide not supplied with this exam or by the instructor is strictly prohibited.

On my honor, I have neither given nor received illicit information during this exam.

Signature:

Problem Point Value Score

1 10

2 23

3 10

4 10

5 12

6 10

Total 75

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1. (10 pts) Find and state the local maximum and minimum values and saddle point(s) of the function f(x,y) = x4 + 2y2 − 4xy.

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2. Let R be the region in the xy-plane bound by the lines x − y = 0, x + y = 6 and y = 0.

(a) (5 pts) Sketch and shade the region R. Make sure your sketch is well labeled.

(b) (6 pts) Describe R with inequalities for the order of integration dy dx.

(c) (6 pts) Describe R with inequalities for the order of integration dx dy.

(d) (6 pts) Evaluate ∫∫ R

1 dA.

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3. (10 pts) Evaluate ∫∫ R

(sec2( π

32 y3)) dA where R is the region bound by x = 0, y = 2 and x = y2.

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4. (10 pts) Evaluate ∫∫ R

(x2 + y2 + 1) dA, where R is the region in the 1st quadrant between the circles

x2 + y2 = 1 and x2 + y2 = 4.

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5. Let E be the solid bound by x + 3y + 2z = 6, x = 0, y = 0, and z = 0.

(a) (6 pts) Express the volume of E as a triple iterated integral using dz dy dx as the order of integration. Do Not Evaluate.

(b) (6 pts) Express the volume of E as a triple iterated integral using dx dz dy as the order of integration. Do Not Evaluate.

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6. (10 pts) Evaluate

π∫ 0

2∫ 0

√ 4−z2∫ 0

z sin (y) dx dz dy

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