HKUST. MATH1014 Calculus II. Directions:

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1 HKUST MATH114 Calculus II Midterm Eamination (Sample Version) Name: Student ID: Lecture Section: Directions: This is a closed book eamination. No Calculator is allowed in this eamination. DO NOT open the eam until instructed to do so. Turn off all phones and pagers, and remove headphones. All electronic devices should be kept in a bag awa from our bod. Write our name, ID number, and Lecture Section in the space provided above, and also in the Multiple Choice Item Answer Sheet. Make sure ou also write and mark our ID number correctl in the I.D. No. Bo, in the Multiple Choice Item Answer Sheet. DO NOT use an of our own scratch paper. Write our name on ever scratch paper supplied b the eamination, and do not take an scratch paper awa after the eamination. Answer all questions. Show an appropriate amount of work for each long problem. If ou do not show enough work, ou will get onl partial credit. When instructed to open the eam, please check that ou have 7 pages of questions in addition to the cover page. There are two blank pages and a formula sheet attached which can be used as scratch paper. You ma write on the backside of the pages, but if ou use the backside, clearl indicate that ou have done so. Cheating is a serious violation of the HKUST Academic Code. Students caught cheating will receive a zero score for the eamination, and will also be subjected to further penalties imposed b the Universit. Please read the following statement and sign our signature. I have neither given nor received an unauthorized aid during this eamination. The answers submitted are m own work. I understand that sanctions will be imposed, if I am found to have violated the Universit s regulations governing academic integrit. Student s Signature : Question No. Points Out of Q Q Q Q Total Points 1

2 1 Part I: Answer all of the following multiple choice questions. Mark our MC answers to the boes in the Multiple Choice Item Answer Sheet provided. Write our name and mark our student ID number on the Multiple Choice Item Answer Sheet. Mark onl one answer for each MC question. Multiple answers entered for each single MC question will result in a point deduction. Write also our MC question answers in the following boes for back up use onl. The grading will be based on the answers ou mark on the MC item answer sheet. Question Total Answer Each of the following MC questions is worth 5 points. No partial credit. 1. What is the color version of our eamination paper? Make sure our ID number has also been marked correctl in the I.D. No. Bo in the MC answer form. If ou do not do both correctl, ou lose the points of this question. (a) Green (b) Orange (c) White (d) Yellow (e) Makeup. Find the area enclosed b the curves defined b the equations = 5 and =. (a) 4 (b) 5 (c) (d) 7 (e) 8

3 . Evaluate the integral 4 sin()cos (4)d. (a) 5 (b) (c) 7 (d) 8 (e) 9 4. Evaluate the integral d. (a) 1π (b) 1π (c) π (d) 4π (e) π 5. Evaluate the integral + (+1)(+) d. (a) ln ln (b) ln ln (c) ln ln (d) 4ln+ln (e) 5ln 4ln

4 . The region enclosed b the graph of the function = 4 and the line = is rotated about the -ais to generate a solid of revolution. Find the average value of the cross section areas of the solid which are perpendicular to the -ais. 8π 7π π 5π 4π (a) (b) (c) (d) (e) 7. The base of a solid sitting on the -plane is the region enclosed b the curves = sin and = 4cos over the interval π π. Supposethat each cross section of the solid perpendicular to the -ais is an equilateral triangle (i.e. all sides have equal length). Find the volume of the solid. (a) 9 π (b) 7 π (c) 5 π (d) π (e) π 8. Let f be a positive differentiable function satisfing f() =, f () =, f(1) = 4, f (1) =. Evaluate the integral 1 f ()ln(f())d. (a) ln 1 (b) ln 1 (c) 4ln (d) 4ln ln (e) ln

5 4 9. The curve defined b =, where 4, is rotated about the -ais to generate a surface of revolution. Find the area of the surface. 49π 5π 8π (a) (b) (c) (d) 9π 14π (e) 1. Find the slope of the tangent line to the curve defined b the polar equation r = +sinθ cos θ at the point on the curve with angular coordinate θ = π 4. (a) (b) 5 (c) 7 4 (d) 9 5 (e) A curve in the -plane is defined b the polar equation r = secθ, where θ π. Which of the following definite integrals is the arc length of the curve? (a) (b) (c) (d) (e) 1+ 1 secθ dθ tanθ+ 1 secθ dθ sec θdθ secθ + 1 tanθ dθ tanθ +sec θdθ

6 5 Part II: Answer each of the following questions. 1. [15 pts] The region enclosed b the curves = ln(1 ) and = ln is rotated about the -ais to generate a solid of revolution. (a) Epress the volume of the solid as a definite integral. [5 pts] (b) Find the volume of the solid. [1 pts]

7 1. [15 pts] A container has the shape of a bowl obtained b rotating the curve = 8, 1, around the -ais. The lowest point of the container is 5 m above the ground. Find the work required to fill the tank b pumping water from the ground through a thin tube connecting the ground to the bottom of the tank. (Ignore the water in the thin tube.) Recall the densit of water is ρ = 1 kg/m, and g = 9.8 m/s is gravit acceleration. [15 pts] (You ma use the smbols ρ and g to represent our answer.) Ground

8 7 ( ) 14. [15 pts] Consider the curve defined b the equation =. 1+ (a) Show that the polar equation of the curve can be written as r = (cos θ sin θ)secθ. [4 pts] (b) Find the area inside the loop of the given curve. (Hint: At what θ will the curve hit the origin?) [7 pts] (c) Use integral(s) to epress the area inside the loop of the given curve but outside the circle given b the polar equation r = cosθ. Do not evaluate the integral. [4 pts] (Hint: At what θ will the curves intersect each other?)

9 Math114 Midterm Eam Formula Sheet Trigonometric Identities cos θ+sin θ = 1 1+tan θ = sec θ 1+cot θ = csc θ sinθ = sinθcosθ cosθ = cos θ 1 = 1 sin θ = cos θ sin θ tanθ = tanθ 1 tan θ sin(a+b) = sinacosb +sinbcosa sin(a B) = sinacosb sinbcosa cos(a+b) = cosacosb sinasinb cos(a B) = cosacosb +sinasinb ( ) sinacosb = 1 sin(a+b)+sin(a B) ( ) cosacosb = 1 cos(a+b)+cos(a B) ( ) sinasinb = 1 cos(a B) cos(a+b) tan(a+b) = tana+tanb 1 tanatanb tan(a B) = tana tanb 1+tanAtanB cos n d = 1 n cosn 1 sin+ n 1 n sin n d = 1 n sinn 1 cos+ n 1 n sec n d = 1 n 1 secn tan+ n n 1 cos n d sin n d sec n d

HKUST. MATH1013 Calculus IB. Directions:

HKUST. MATH1013 Calculus IB. Directions: HKUST MATH101 Calculus IB Midterm Eamination (Sample Version) Name: Student ID: Lecture Section: Directions: This is a closed book eamination. No Calculator is allowed in this eamination. DO NOT open the