NAME: (print!) Dr. Z. RUID: (print!) None SSC: (circle) None / I / II / I and II

MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. , Final Exam , Tue., Dec. 15, 2020 Email the completed test, renamed as finalFirstLast.pdf to DrZcalc3@gmail.com no later than 3:30pm, (or, in case of conflict, three and half hours after the start).

WRITE YOUR FINAL ANSWERS BELOW

1. −18

2. ∫0^{\frac{1}{2}} ∫ 1_{\frac{1}{4}} f(x, y) dx dy + ∫ 1_{\frac{1}{2}} ∫_{y^{2}}^{1} f(x, y) dx dy +

3. z = −\frac{1}{2}x − \frac{5}{6}y + \frac{7}{18}π

4. 3i − 6j + 9k OR < langle3, −6, 9⟩ >

5. \frac{π}{3} , \frac{π}{3} , \frac{π}{3}

6. 3 \sqrt{3}

7. 6 cos 2 OR −2.496881019

8. 8π

9. −15

10. (\frac{3}{4}, −1), saddle point.

11. 3\frac{1}{3000} OR \frac{9001}{3000} OR 3.0003333 ….

12. \frac{\sqrt{2}}{6}

13. ∫_{0}^{2} ∫ π_{\frac{π}{2}} ∫0^{\frac{π}{2}} ρ^{6} sin^{4} ϕ sin^{2} θ cos^{2} θ dρdθdϕ

14. \frac{1}{3}

15. ∫_{0}^{1} ∫_{0}^{u} 2\sqrt{u^{4} + 4u^{2}v^{2} + v^{4}} dv du

16. 14

17. 0 Sign the following declaration:

I Hereby declare that all the work was done by myself. I was allowed to use Maple (unless specifically told not to), calculators, the book, and all the material in the web-page of this class but not other resources on the internet.

I only spent (at most) 3 hours on doing the exam. The last 30 minutes were spent in checking and double-checking the answers.

I also understand that I may be subject to a random short chat to verify that I actually did it all by myself.

Signed:

Possibly useful facts from school Geometry (that you are welcome to use) : (i) The area of a circle radius r is πr^{2}. (ii) The circumference of a circle radius r is 2πr (iii) The parametric equation of an ellipse with axes a b and parallel to the x and y axes respectively is x = a cos θ, y = b cos θ, 0 < θ < 2π. (iv) The area of an ellipse with axes a and b is πab (v) The volume and surface area of a sphere radius R are \frac{4}{3}πR^{3} and 4πR^{2} respectively (vi) The volume of a cone is the area of the base times the height over 3. (vii) The volume of a pyramid is the area of the base times the height over 3. (viii) The area of a triangle is base times height over 2.

Formula that you may (or may not) need

If the surface S is given in explicit notation z = g(x, y), above the region of the xy-plane , D, then

∫ ∫_{S} F · dS =

∫ ∫_{D} (−P \frac{∂g}{∂x} − Q\frac{∂g}{∂y} + R) dA .

1. (12 pts.) Without using Maple (or any software) Compute the vector-field line integral

∫_{C} (cos (e^{sin x}) + 5y) dx + (sin (e^{cos y}) + 11x) dy ,

over the path consisiting of the five line segments (in that order)

(1, 0) → (−1, 0) → (−1, 1) → (0, 2) → (1, 1) → (1, 0) .

Explain!

ans. −18

Since this is a closed path, we should use Green’s Theorem. Q_{x} − P_{y} = 11 − 5 = 6 and we need to integrate it over the region inside the curve. Since the integrand is constant, i.e. 6, the area-integral is simply 6 times the area. This is a ”house” where the roof is the triangle with vertices (−1, 1), (1, 1), (0, 2) whose area is 2 · 1/2 = 1, and the main part of the ”house” is the rectangle with vertices (1, 0), (−1, 0), (−1, 1), (1, 1) whose area is 2 · 1 =, so the total area of the region is 3. So Green’s theorem tells you that the answer is 18. But notice the orientation it is clockwise, so we have to multiply by −1, getting that the final answer is −18.