xukun235
幼苗
共回答了19个问题采纳率:94.7% 举报
1.r^3×√(1+r^2)dr=1/2×r^2×√(1+r^2)d(1+r^2)
设t=√(1+r^2),则1+r^2=t^2,r^2=t^2-1,所以
∫r^3×√(1+r^2)dr
=1/2×∫(t^2-1)×t×d(t^2)
=1/2×∫(t^2-1)×t×2tdt
=∫(t^4-t^2)dt
=1/5×t^5-1/3×t^3+C
=1/5×√(1+r^2)^5-1/3×√(1+r^2)^3+C
=1/15×√(1+r^2)^3×(3r^2-2)+C
2.
曲面向yz面投影,把曲面分为前后两部分:
S1:x=√(4-y^2)
S2:x=-√(4-y^2)
S1与S2在yz面上的投影都是矩形区域:0≤z≤3,-2≤y≤2
dS=2/√(4-y^2)dydz
所以,
∫∫z^2 dS
=2∫∫z^2×2/√(4-y^2) dydz
=4∫0~3 z^2 dz ∫`-2~2 1/√(4-y^2) dy
=4×9×π=36π
1年前
2