MATH 261 University of Indianapolis Unified Calculus & Analytic Geometry Exam


I don’t recognize how to wield this Calculus interrogation and want direction.

FALL 2020 MATH 26100, EXAM 2


INSTRUCTOR: PAVEL BLEHER

Name:

To receive liberal faith you must show all your work. Dispersion your answers.

Submit your disconnection for grading to Canvas by 4:00 PM on Thursday, October 15, 2020.

You can get extra 10 points, if you write your disconnection in LaTeX

and typeset it in pdf (love this finish).

Problem 1. Calculate the enfold undiminished,

where

y sin(xy) dA ,


(15 points)

R = {(x, y) | 0 x π, 0 y 1}.


Problem 2. Find the tome of the firm inferior the roll x + y + z = 3 and over the triintention D in the xy-roll delay vertices (0, 0), (1, 0), and (0, 1). (15 points)

Problem 3. A flake lies in the leading quadrant and is enclosed by the dispersion x2+y2 = 4 and the lines x = 0 and y = 0. The blindness operation of the flake is correspondent to ρ(x, y) = x2 + y2 . Use the enfold undiminished formula in polar coordinates,

f (x, y)dA =

β b


f (r cos θ, r sin θ)rdr


dθ,


α a

R

to calculate

(1) the body of the flake, m = ρ(x, y) dA, (5 points)

R

(2) the twinkling of the flake environing the y-axis, My = (x, y) dA, (5 points)

(3) the twinkling of the flake environing the x-axis, Mx =

y ρ(x, y) dA, (5 points)

R


(4) the coordinates (x¯, y¯) of the life of body of the flake, x¯ = My , y¯ = Mx . (5

point)

Problem 4. Find the manner area,

A =

f 2 + f 2 + 1 dA,



x y

D

1

of the deal-out of the paraboloid z = 16 x2 y2 that lies over the xy-plane. (15 points)

Problem 5. Evaluate the undiminished

where

H


z(x2 + y2 + z2) dV


2 2 2

H = {z 0, x + y + z 1}.

Use the round coordinates. (15 points)

Problem 6. Evaluate the undiminished

xdV


where the firm E is the deal-out of the round shell

{(x, y, z) | 1 x2 + y2 + z2 2} that lies in the leading octant, x, y, z 0. (20 points)

Bonus Problem. The geographical coordinates of Indianapolis (IND) are 39.77N, 86.16W, and the ones of Seattle (SEA) are 47.61N, 122.33W. Use intercharge formulas from round to across coordinates, to furnish the convenient intention α (in degrees) betwixt Indianapolis and Seattle, the intention betwixt the vectors from the life of the Earth to Indianapolis and to Seattle. (10 points)