**Exam 3 (MTH 252)-Spring 2023**

加拿大微积分Final exam代考 Determine whether or not the following vector fifield is conservative. Explain your rea-soning well to support your claims.

Exam 3 will be on April 19 and covers Sections 15.3-15.9 (except 15.4) and 16.1-16.3.You should review your class notes and examples discussed during our meetings. I will hold a review session for the exam on Tuesday from 4:00-5:30 pm in BAC 245. Below are some problems to practice for the exam. We will discuss some of these problems during the review session.

### 1.Evaluate each integral:

(a) *A *with *R *= *{*(*x, y*)*| **x*^{2} + *y*^{2} *≤ *4*, x **≥ *0*, y **≥ *0*}*.

(b)

2.Find the volume of the following solids.

(a) The solid bounded by the paraboloids *z *= *x ^{2}* +

*y*and

^{2}*z*= 2

*−*

*x*

^{2}*−*

*y*.

^{2}(b) The solid bounded by the cylinder *x ^{2}* +

*y*= 4 and

^{2}*z*= 3

*−*

*x*and

*z*=

*x*

*−*3.

3.Find the area of the part of the sphere *x ^{2}* +

*y*+

^{2}*z*= 4 that lies above the plane

^{2}*z*= 1.

### 4.Rewrite the integral 加拿大微积分Final exam代考

as an iterated integral in the order *dydzdx*.

5.Rewrite the integral

as an iterated integral in the order *dy dz dx *and *dx dy dz*.

6.Evaluate each integral:

(a) where *E *is bounded by the paraboloid *x *= 4*y ^{2}* + 4

*z*and the plane

^{2}*x*= 4.

(b) * *where *E *is enclosed by the surfaces *z *= *x ^{2}*

*−*1 and

*z*= 1

*−*

*x*,

^{2}*y*= 0,

*y*= 2.

### 7.Find the volume of the solid that is enclosed by the cone *z *= p *x*^{2} + *y*^{2} and the sphere 加拿大微积分Final exam代考

^{2}

^{2}

*x ^{2}* +

*y*+

^{2}*z*= 2.

^{2}8.Evaluate where *E *is the solid that lies within both the cylinder *x ^{2}*+

*y*= 1 and

^{2}*x*+

^{2}*y*= 16 above the

^{2}*xy*-plane and below the plane *z *= *y *+ 4.

9.Evaluate where *E *is the solid given by *x ^{2}* +

*y*+

^{2}*z*

^{2}*≤*9 and

*x*

*≥*0.

### 10.Evaluate the integral where *R *is the region with vertices (1*, *0), (2*, *0),(0*, *2), and (0*, *1).

11.Evaluate the integral where *R *is the region with vertices (0*, *0),(1*, **−*1), (2*, *0), and (1*, *1).

12.Evaluate the integral where *R *is the region given by the inequality *|**x**|*+*|**y**| ≤1.*

### 13.Find the volume of the solid outside the cone φ* *= *π/*4 and inside the sphere *ρ *= 4 cos φ. 加拿大微积分Final exam代考

14.Find the volume of the part of *ρ **≤ *2 that lies between the cones φ* *= *π/*3 and φ* *= 2*π/*3.

15.Show that the line integral where and *C *is any path from (1*, *0) to (2*, *1), is independent of path and then evaluate the integral.

### 16.Determine whether or not the following vector fifield is conservative. Explain your rea-soning well to support your claims.

17.Determine wether or not the vector fifield is conservative. If it is conservative, fifind a function *f *such that * *= *∇**f*.

18.Assume (*x, y*) = 〈* *3*x*^{2} + *y*^{2} *, *2*xy〉** *and *C *is the curve shown below: 加拿大微积分Final exam代考

(a) Evaluate∫_{C}**⋅***dr*.

(b) Show that is conservative and fifind a function such that * *= *∇**f*.

(c) Evaluate ∫_{C}* **· **dr *using the Fundamental Theorem of Calculus for line integrals.

(d) Evaluate ∫_{C}* **· **dr *by fifirst replacing *C *by a simpler curve that has the same initial and terminal points.

### 19.Evaluate R *C *(*x *^{2}+*y *^{2} +*z *^{2} )*ds*, where

^{2}

*C *is the curve with parametric representation (*t*) =*〈 **t, *cos 2*t,*sin 2*t*〉and 0*≤ **t **≤ *2*π*.

20.The fifigure shows a vector fifield *F~ *and two curves *C*1 and *C*2. Are the line integrals of over *C*_{1 }and *C*_{2 }positive, negative, or zero? Explain.

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