. Find the volume and centrcid of the ice-cream cone

In Problems 1 through 5, evaluate the given integral by first revirsing the ordler of integration.

In Problems 1 through 5, evaluate the given integral by first revirsing the ordler of integration.

In Problems 1 through 5, evaluate the given integral by first revirsing the ordler of integration.

In Problems 1 through 5, evaluate the given integral by first revirsing the ordler of integration.

In Problems 1 through 5, evaluate the given integral by first revirsing the ordler of integration.

The double integral I I - r/r',/.r' J,, J. \ is an improper integral over the unbounded region in the lirst quadrant between the lines -r' : .r and.r : 0. Assunling that it is valid (it is) to reverse the order ol integration. evaluate this integral by integrating first with respect to.\.

Find the volume of the soiid f that lies below the paraboloid : : ,rl + r,r and above the rriangle R in rhe n'plane that has Yenices at (0.0.0). ( I. l. 0). and (2.0.0).

Find by integration in cylindrical coordinates the volume bounded by the paraboloids : : 2.rr + 2r'r and: : ,18

Use integration in spherical coordinates to find the volume and centroid ol the solid legion that is inside the sphere p : 3, below the cone d : z/3, and above the.r-r'plane Q:tt/?.

Find the volunre of the solid bounded by the elliptic paraboloids : = -r: + 3r'l and : = 8 - i: - -51:

Find the volume bounded by the paraboloid l = .tr + 3:r and the parabolic cylinder -r' : 4 - :r.

Find the volume of the region bounded by the parabolic cylinders: = -tl, : = 2-.tl and the planes r' : 0, .r'*r : .1.

Find the volume of the region bounded by the elliptical cylinder r i - ,l:l :4 and the plane:.r - O..\ = _r + l.

Show that the volume of the solid bounded by the elliptical cylinder ;-;-l n- Dand the planes: = 0.: =,/, +.r (where,lr > a > 0) is V = T ablt.

Lel R be the first-quadrant region bounded by the curve .rr + rlt : : r'l and the line ,t = r. Use polar coordinates to evaluate tt r // '"R ,r* i- llla

The region bounded by l = .rr and r = r'r: 6(r.r') : ..: + .r'l

The region bounded by,r : 2,r'r and r': : -r - 4: 6(.r. r') = j-

The region between r' = ln-r and the,t-axis over the interval llr !2: 6(.t. r') = l/.r'

The circle bounded bv r' : 2 cos d: 6 (r'. 0) = ,t (3 senstanll

The legion of Problem 19: d(r. 0) : r

, Use the iirst theorem of Pappus to lind the r,-coordinate of rhe centroid of the upper half of the ellipse .Ll i: - I -1. a: b: Employ the facts that the area of this semiellipse is A = r.rlrl2 and the volume of the ellipsoid it generates when rcrued around rhe.r_ax1s 15 y : ]zaD:

(a) Use the nrst theorem of Pappus to find the centroid of the first quadrant ponion oi the annular ring with boundary circles.rl +r'l - arl and tl *.r'l : b: (where0 < a < 6). (b) Show thal the limiting position of this centroid as b -> rr is the cenlroid of a quarter-circular arc. as we found in Probiem,l,l of Section 13.5.

Find the centroid of the region in the 5-plane bounded by the .r-axis and the parabola r' : -l - .t:.

Find the volume of the solid that lies below the paraboiic cylinder : = 12 and above the triangle in the -r,r-plane bounded by the coordinate axes and the line ,r- + r' = l.

Use cylindrical coordinates to Rnd the volume of the icecream cone bounded above by the sphere,rt + ): +;t : 5 and belon by the cone : = 2F-l!

. Find the volume and centrcid of the ice-cream cone bounded above by the sphere p = a and below by the cone O = n 13

A homogeneous solid circular cone has mass M and base radius a. Find its moment of inertia arcund its axis of svmmetry.

Find the mass of the iirst octant of the ball p ! a if its densily at Lr. r. z ) i\ 6 tr. r'. :) : .r'\:

Find the moment of inenia around the r-axis of the homogeneous solid ellipsoid with unit density and boundary surf'ace r.r t- -_- _ _:_ r :, _ l

Find the volume of the region in the first octant rhar is bounded by the sphere p = ./. the cylinder,. = a. the plane : : a. the.r:-plane, and the r,;-plane

Find the moment of inenia around the --axis of the homogeneous region of unit density that Iies inside both rhe sphere p = 2 and the cyiinderr = 2cos6.

Find the volume of the solid obtained by revolving around the )-axis the region inside the circle r : 2a cos 0.

Find the volume of the solid obtained by revolving around the .jr-axis the region enclosed by the cardioid r : I * cos 6.

Find the volume of the solid torus obtained by revolving the disk 0 ! i' ! a around the iine,r : -b. )b)Z a.

Assume that the to.us of Problem 34 has uniform density 6. Find its moment of inenia around its natural axis of symmetry.

Show thal the average distance o[ the pninrs ota disk of radiu\ d from irs center is :d/-1.

Show that the average distance of the points of a disk of radius (l from a fixed point on its boundary is 32al9r.

A circle of radius I is interior to and tangent to a circle of radius 2. Find the average distance of the point of Iangency from the points that lie between the lwo circles

Show that the average distance of the points of a spherical ball ofradius d from its center is 3a/4.

Show that the average distance of the points of a spherical ball of radius a from a nxed point on its surface is 6d15.

A sphere of radius I is interior to and tangent to a sphere of radius 2. Find the average distance of the point of tangency from the set of all points between the two spheres,

A right circular cone has radius R and height 11. Find the average distance ofpoints of the cone from its venex.

A right circular cone has radius R and height 11. Find the average distance ofpoints of the cone from its venex.

Find the surface area of the pan of the sudace a = J'r - r: that is inside the cylinder -r2 + r'l = 4

Let A be the suface area of the zone on the sphere p : d between the planes: : :l and z : a2 (where -ds;l < ::5 a). Use the formula of Problem 18inSection l3.8to show that A : 2rralt-where lt : :l - at.

Find the surface area of the pan of the sphere p = 2 that is inside the cylinder,r: a -r': : 2-r.

A square hole \\'ith side length 2 is cut through a cone of height 2 and base radius 2: the centerline of the hole is the axis oi symmetry of the cone. Find the area of the surface removed from the cone

Numerical)y approximate the surface area ol the part of the parabolic cl'linder 2: - -r: that lies inside the cylinder ,rl+r'2:1.

A "fence" of Variable height h0) stands above the plane curve (r (t). ,] (r.)). Thus the fence has the parametrization .r = jr(t), l :,](I).; = ; for a:t:b.0=:.:h(r). Apply Eq. (8) of Section 13.8 ro show rhat the area of the fence is

Apply the formula of Problem .19 to compute rhe area of the pan ofthe cylinder r : asind lhat lies inside the sphere al+-:l:al.

Find the polar moment of inertia ol the lirst-quadrant region ol constant densily 6 that is bounded by the hyperbolas ,rr' = I,'l.\. : 3 and.r-: - r.l = l.rl - r.l = 4.

Substitute rr :.\- - 1, and r) :.r + ] to evaluate ll /r'-1\ / / exp( - jr/.r rir. ""n where R is bounded by the coordinate axes and rhe line

Use el)ipsoidal coordinates .r - .tpsindcos0. -1. = bp sin @ sin e. : = cp cos d to find the mass of the solid ellipsoid .1-l tl -:l - - ,; - ; 1 | (I- Dif its densitl' at the point ("r. 1 . :) is given by ^ = I": 1""',1(#i * (f)] " n.,, ,r'_ -l - :-

. Let R be the firsGquadrant rcgion bounded by the lemniscates 12 : 3cos20, 12 = 4cos20 and 12 = 3sin29. 12 =4sin2g (Fig. I3.MPl). Show rhat its area is . 10_74 " = ___ 4__. lSuggestion: Denne the tmnsformation I from the ay-plane tother6-planeby/2:ut/2cos20,r2 = url2 sin 29. Show first that . ull r.=_. e= u+v Then show that 0(r.0) I : --- l Afu, v) l6rtu + vt3/2 '

A 2-by-2 square hole is cur symmelrically rhrough a sphere of radius r,/3. {See Fig. l3.MP2.) (a) Show thar rhe loral surface area of the two pieces cut from the sphere is ft - t | \ I = I AJI arcsinl - Jo \.,/3 =)o' Then use Simpson's rule to approximate this integnl. (b) (Difficuh!) Show that rhe exact value of the integral in par (a) is A = 4tr (JT - D. [suggestion: First inreg;te by parts, then substirute x = J2 sin?.l

Show that the volume enclosed by the surface r1l3 + ),2/3 +,1/3 _ a2/3 ls V = |r a3. lsuggestion: Substitute y = , sin3 d.l

Show that the volume enclosed by the surface lrlr 3 + lll' J + lrlr r =4r'r is V: *Ia3. [suggestion: Substitute ) = , sin6 6.]

Find the average of the squarc of the distance of points of rhe solid ellipsoid (x /a)2 + (j/b)2 + (z/c)2 S I from the origin

A cube C of edge length 1 is rotated around a line passing through two opposite vertices, rhereby sweeping out a solid S of revolution. Find the volurne of S. (Answe,r: n/./3 * L8 138.)