(Solved): (Section 10.2) Consider the R^(2)-R function defined by f(x,y)=(y)/(x)-lnx+(1)/(2)y^(2) Find the cri ...
(Section 10.2) Consider the
R^(2)-Rfunction defined by
f(x,y)=(y)/(x)-lnx+(1)/(2)y^(2)Find the critical points of
fand their nature. [10] 2. (Section 10.3) A delivery company needs the measurements of a rectangular box such that the length plus twice the width plus the height should not exceed 300 cm . Use the method of Lagrange to find the maximum volume of such a box? 3. (Sections 13.1-13.5) Let
Rbe the triangle in
R^(2)with vertices
(1,1),(2,3)and
(4,2). (a) Sketch the region
R. Find the points of intersection of the line and the circle and indicate them on your sketch. (b) Is
Ra Type I region? If yes, describe it as a Type I region. If not, describe it as a union of Type I regions. Use set builder notation. (c) Is
Ra Type II region? If yes, describe it as a Type II region. If not, describe it as a union of Type II regions. Use set builder notation. (d) Calculate the integral
∬_(R)x+ydAi. by using the order of integration
dydxii. by using the order of integration
dxdy. [18] 4. (Sections 3.2,13.3,13.7,14.3,14.6,14.7) Consider the integral
\int_(-1)^1 \int_0^(\sqrt(1-x^(2))) 1-y^(2)dydx(a) Sketch the region of integration. (3) (b) Give a geometric interpretation of the above integral by using a 3-dimensional sketch. (4) (c) Transform the above integral to a double integral with polar coordinates (Do not evaluate the integral). [11] Evaluate the line integral
\int_C xyds,where
Cis given by the line
y=lnx,1<=x<=e. [7] 6. (Chapter 15, Section 16.5) A particle moves from
(1,1,0)to
(-1,1,-1)under the influence of the vector field
F_(x,y)=(2xyz,x^(2)z+2yz^(2),x^(2)y+2y^(2)z+e^(z)).(a) Show that the integral is independent of the path followed by using the Fundamental Theorem of Line Integrals (Theorem 16.5.1). (b) Compute the integral i. by transforming the integral to an ordinary single integral (see Section 15.3) ii. using the Fundamental Theorem of Line Integrals. [11] [TOTAL: 67]
