Midterm 3 Review Solutions by CC

Transcription

1 Midterm Review Solution by CC Problem Set u (but do not evaluate) the iterated integral to rereent each of the following. (a) The volume of the olid encloed by the arabaloid z x + y and the lane z, x : Clearly, the range for the x coordinate i given by ; a it mut lie between the the lane. The z coordinate i beteween and f (x; y) x + y. We oberve that for jxj < and y mall, f (x; y) < o that f (x; y) hould be the lower limit, and the uer limit. Finally, we nd the boundary of the region for the y coordinate. Thi occur when the two urface interect, i.e. x + y ) y x x x! dz dy dx x +y (b) The volume of the olid beneath the urface z xy and above the triangle having coordinate (; ), (; ), and (; ). The z coordinate will range from to xy. The range for the x and y coordinate in the region bounding the given triangle in the xy lane. Note that the x coordinate range from to, and the y coordinate can be bounded between x and + x for x, and between x and + x for x. Therefore the volume can be exreed a + x x xy dzdydx + + x x xy dxdydx (c) The average value of the function z (x + y) over the rectangle [; ] [; ] : We want to integrate the function over the given rectangle, then average it. Thi i given by R R (x + y) dydx R dydx 4 R (x + y) dydx (d) The area of the region outide r co and inide r + co. Since the curve are given in olar coordinate, we will tay in that ytem and exre the integral in that manner. Firt, note that the two curve interect when co + co, i.e. when co or co and alo at. By ymmetry, it u ce to conider the region between and, and then multily it by a factor of. Therefore, the area between them will be given by ( +co rdr! d co rdr! d )

2 Below, r + co i ictured in red and r co in black. y x. (e) The volume of the olid between the cone z x + y and the here x + y + z 4. To eae notation, it make ene to olve thi roblem in cylindrical or herical coordinate rather than rectangular. We do o in cylindrical coordinate. Rewriting, the cone i then given by and here by z r z 4 r : There will be two halve to thi volume, ymmetric about the xy lane. We thu (equivalently) write the exreion for twice the volume of the to half. The z coordinate will range from r to 4 r (above the cone, beneath the here). The radial coordinate can range from (at the origin) to at mot the outer boundary when the here and cone meet, i.e. when r 4 r ) r : Finally, the coordinate i unretricted ince we have ymmetry. integral can be written a!! r 4 r rdz dr d (f) The area of the ortion of the hyerbolic araboloiod z x the cylinder x + y 4 and x + z 9: Thu, the y between

3 We want to comute the urface area in the annulu between x + y 4 and x + y 9. For the given urface z x y, y o x + y da () We could write out the limit of integration in rectangular coordinate, but it cleaner to witch to olar. Then the boundarie are given by r and r and () yield A r + r drd (g) The area of the art of the here x + y + z 4 that lie above the lane z y. We will break thi roblem into three art, two from the to art of the here and one from the bottom art. In the lane y >, the relevant area i jut that of the to half of the here that lie above y, and in the lane y < we mut account for the entire to and ome of the bottom. We will write noting z to + y z bot x y to y x y bot y y The interection of the lane and the here occur when y y y 4 x y r y

4 Then the contribution from the y > half above the xy lane i A x + y y da q + x + y y dydx r +4 in + r 4 r rdrd and the contribution from y < above the xy lane by + + x + y y da r + r 4 r da ince the domain i the whole emicircle. Finally, the art below the xy-lane but above z y i given (by ymmetry) a A, and the reult i A A + A : (h) The volume bounded by the cone z x +y ; the cylinder x +y 6, and the lane z and z 4: We will olve thi roblem in cylindrical coordinate for convenience. Firt, we rewrite the equation of the cone and cylinder a z r, r 4; reectively. We will calculate the art < z < 4 and The former i rdrdzd and the latter a V 4 4 z 4 z rdrdzd rdrdzd < z < earately. 4

5 Problem Calculate the volume of the olid encloed by the urface z + x ye y and the lane z ; x, y, and y. The domain i given by < x < and < y <, o the volume underneath the grah can be exreed a x + x + x ye y dydx y + x ye y x e y y y dx + x e x e + x dx x x + 8 Problem Suoe you have a olid ball of radiu. Find the average ditance between a oint in the ball and the center of the ball. We are aked to nd the average ditance between a oint in a ball and it center. Center the ball at the origin for imlicity, and ue herical coordinate. The ditance from an arbitrary oint (; ; ) to the center (; ; ) i imly. The average ditance i then integrated over the here, normalized by the total volume of the here. A we know the volume of a here of radiu R i 4 R, we then have 4 d () in ddd S in ddd 4 (4) Problem 7 Calculate R R x + y da where i the region in the rt quadrant of the xy lane bounded by y, y x; and x + y 9: It make ene to tranform to olar coordinate to evaluate thi roblem due to the tructure of the integrand. We rt rewrite the bound on the region given in term of olar coordinate: y ) y x ) tan x + y 9 ) r :

6 Thi i a ector of a circle, in articular, the ection from to of the circle of radiu. We now erform the integration: x + y da 4 r r rdrd r d r 8. Problem 9 Evaluate R F ~ x d~r, where F C ; y and C i obtained by x +y x +y following the arabola y + x from the oint ( ; ) to the oint (; ). Method. Oberve that F i a conervative eld, and in fact, for x x + y! xy x f : x + y ;! y x + y we have r f ~ F ~. Thu, by the fundamental theorem for line integral, ~F d~r fj (;) ( ;) + q( ) + : C Method. We can alo olve thi roblem by traight comutation, arametrizing the curve by x (t) t ; y (t) + (t ) t t + for t. Then we have ~F (~r t t t + ; q A q + (t ) + (t ) and o that C ~F d~r ; dx dt; dy t q + (t ) A () + ) q t t + A (t ) A dt + (t ) 6

7 matching our anwer from method. Problem etermine if each of the following vector eld i conervative. If o, nd a function f uch that ~ F r ~ f. (a) ~ F (x + y)^{ + x + y ^ Set P x + y; Q x + y. We then ; imlying the vector eld i conervative. To comute f, we note that f P dx (x + y) dx x + yx + g (y) where g i a function of y but not x. We di erentiate f to nd g: f y x + g (y) Q x + y and conclude g (y) y + K for a contant K. Therefore f x + yx + y + K; K R. To check, we can di erentiate and verify f x x + y P f y x + y Q (b) ~ F (x + y)^{ + x + y ^ Set P x + y; Q x + y :We note but that Since : we conclude the vector eld ~ F i not conervative. (c) F ~ (x co y + )^{ + ( x in y + x) ^ Set P x co y + ; Q x in y + x: We have x in y; in y + : we conclude the vector eld ~ F i not conervative. 7

8 (d) ~ F x co x + y x in x + y ^{ x y in x + y ^ Set P x co x + y x in x + y ; Q x y in x + y : Then we have 4xy in x + y yx in x + 4xy in x + y x y co x + y ; which are equal, imlying that the vector eld i conervative. To comute f, we note f P dx x co x + y x in x + y dx x co x + y + g (y) where g i a function of y but not x. To nd g we di erentiate with reect to y: f y x y co x + y + g (y) Q x y co x + y and conclude g (y) K for ome contant K. Therefore f x co x + y + K for K R. To check, we di erentiate and verify f x x co x + y f y yx in x + y Q x in x + y P Problem R Suoe F ~ i a conervative vector eld and C i a cloed curve. Prove that F ~ d~r. You can ue Green Theorem. C Note: There are a number of other condition required for the alication of Green Theorem. We are auming that thee other condition hold. Write F ~ P^{ + Q^. Since F ~ i a conervative vector eld, we have : Furthermore, by Green Theorem, for a cloed curve C: ~F d~r da where i the region encloed by the curve C. 8