# 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 *x*^{2}+*y*^{2} __= 4 an__d the lines *x *= 0 and *y *= 0. The blindness operation of the flake is correspondent to *ρ*(*x, y*) = *x*2 + *y*2 . 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, *M _{y} *=

*xρ*(

*x, y*)

*dA,*(5 points)

(3) the twinkling of the flake environing the *x*-axis, *M _{x} *=

__ __ __ __

*y ρ*(*x, y*) *dA, *(5 points)

*R*

(4) the coordinates (*x*¯*,** **y*¯) of the life of body of the flake, *x*¯ = ^{M}*y ** **, ** **y*¯ = ^{M}*x ** **. ** *(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 *−**x*^{2} *−**y*^{2} that lies over the *xy*-plane. (15 points)

**Problem 5. **Evaluate the undiminished

where

*H*

*z*(*x*^{2} + *y*^{2} + *z*^{2}) *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 *≤ **x*2 + *y*2 + *z*2 *≤ *2*} *that lies in the leading octant, *x, y, z **≥ *0. (20 points)

**Bonus Problem. **The geographical coordinates of Indianapolis (IND) are 39*.*77* ^{◦}*N, 86

*.*16

*W, and the ones of Seattle (SEA) are 47*

^{◦}*.*61

*N, 122*

^{◦}*.*33

*W. 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)