Math 21a Homework 07 Solutions Spring, 2014

Transcription

1 Math a Homework 7 Solutions Spring, 4. valuate the iterated integral. a) Stewart.7 # 6 ) e d d d We perform the iterated integral: e d d d e d d e d [ e [ ] 4 e + 4e. Note that we ve twice done an integral like ] 4 e d e + C e d d ) e [ ] e + 4 d [ e + ) using the substitution u. We did this for both the integral and the integral.) b) Stewart.7 # 8 ) Again we integrate: sin d d d sin d d d This could also be written as π valuate the triple integral: a) Stewart.7 # ) cos d d [ ] sin) d [ cos ) ] π π 4 + ) π 4. cos)) d d ) sin ) d [ 4 π) 4 + cos π) ) ) ) e d ) ] e + 3 sin ) ) d 4 )4 + )] cos ) cos 5 ) dv, where {,, ),, }. B the given description of, we can write the triple integral as an iterated integral: cos 5 ) dv cos 5 ) d d d. This we integrate in the usual wa: cos 5 ) d d d cos 5 ) d d cos 5 ) d d cos 5 ) d 5 sin5 ) ) sin) sin) 5 cos 5 ) 4 d d cos 5 ) d sin). 5 4 cos 5 ) d

2 b) Stewart.7 # 4 ) and +. dv, where is bounded b the parabolic clinders, and the planes One eas wa to describe the region is as {,, ) : +,, ) }, where is the region in the -plane bounded b the two parabolas and. This region we can draw: Thus we can write our integral as either dv + d da + d d d or + d d d. These last two iterated integrals are clearl equivalent because of the smmetr in and ), and we ll compute onl the first: dv [ d d d + d d ] + ) d d d / 7) d [ / 7/ ] [ 4 ) + 3 [ ] ) d d 3/ 6)] d Based on Stewart.7# 4 ) Let be the tetrahedron bounded b the planes, and + +. a) Use a triple integral to find the volume of. Here this tetrahedron is the volume between and over the region in the -plane bounded b the positive aes and the line, or + : +

3 This volume is then given b integrating the constant over this solid : Volume of dv d d d Thus the volume of is /6. ) d d [ ) ] d 6 )3 6 ) 6. ) d b) Suppose that the densit of the tetrahedron at an point,, ) is given b ρ,, ) in lbs/in 3 ). Find the total mass of the tetrahedron. The mass of is given b integrating the densit function: Mass of ρ,, ) dv d d d Thus the mass of is /4 lbs. ) d d [ ) 3 3 ] d 6 4 )4 4 ) In this problem we practice with switching the order of triple integration: ) ) d d ) 3 d a) Stewart.7 #36 ) Write five other iterated integrals that are equal to the given iterated integral: f,, ) d d d. We switch the order of integration in pairs, starting with d d d: The original outer limits for d d gave us the region in the -plane. We write this outer pair of integrals as d d b re-writing as. We get f,, ) d d d. ) Now we do an eample of switching the order of integration for the inner two integrals. In ), the inner two integrals are evaluated over a region that depends on the value of. This region is

4 and so the triple integral can be written as f,, ) d d d. ) Now let s switch the outer integrals, aiming for the order d d d. This involves the region in the -plane, and we get the triple integral f,, ) d d d. 3) To get the order d d d, we ll switch the inner integrals. For fied, the region in the -plane is This gives us the triple integral f,, ) d d d. 4) Finall, the last order is d d d, which we get b switching the outer integrals in 4). This gives us the region

5 in the -plane, which in turn gives us the integral f,, ) d d d. 5) b) Optional: For 4 points tra Credit Stewart.7 #34 ) The figure on page 88 in the tet shows the region of integration for the triple integral f,, ) d d d. Rewrite the integral in the five other orders. We do this as the previous part: switching in pairs, with a sketch for each switch. We otherwise proceed without comment: f,, ) d d d f,, ) d d d f,, ) d d d f,, ) d d d + f,, ) d d d To get to d d d and d d d, we start again with the initial integral: f,, ) d d d f,, ) d d d f,, ) d d d + f,, ) d d d

6 5. Stewart.7 #5 ) The average value of a function f,, ) over a solid region is defined to be f ave f,, ) dv V ) where V ) is the volume of. Find the average value of the function f,, ) + over the region enclosed b the paraboloid and the plane. One eas wa to describe the region is as the set of points between and over the disk given b + in the -plane. Thus and V ) f,, ) dv d da + ) d da Both the resulting double integrals cr out for polar coordinates: V ) ) π da ) da r ) r dr dθ π. + ) ) da. and f,, ) dv + ) ) da π r r ) r dr dθ π 4. This last integral is easiest with the substitution u r, so du r dr and r u. So the r-integral becomes In an case, the average value of f is r r ) r dr u)u du f ave V ) f,, ) dv π/ π 4. u u 3 ) du 4.