计算抛物线y=x²与直线y=2x所围成的面积 求抛物线y=x2与直线y=x,y=2x所围成的图形的面积
\u8ba1\u7b97\u7531\u629b\u7269\u7ebfy=x^2\u4e0e\u76f4\u7ebfy= x\uff0cy=2x\u6240\u56f4\u56fe\u5f62\u7684\u9762\u79ef\u5148\u8ba1\u7b97y=x²\u4e0ey=2x\u6240\u56f4\u6210\u7684\u9762\u79ef
\u8ba1\u7b97y=x²\u4e0ey=2x\u7684\u4ea4\u70b9\uff0c\u5373y=2x=x²\uff0c\u89e3\u65b9\u7a0b\u5f97\u4e24\u4ea4\u70b9\u4e3a\uff080\uff0c0\uff09\u548c\uff082\uff0c4\uff09
\u2234 S1=\u222b\uff080\uff0c2\uff09\uff082x-x²\uff09dx=\uff08x²-1/3*x³\uff09\u4e28\uff080\uff0c2\uff09=\uff084-8/3\uff09-0=4/3
\u518d\u8ba1\u7b97y=x\u4e0ey=x²\u6240\u56f4\u6210\u7684\u9762\u79ef
\u8ba1\u7b97y=x²\u4e0ey=x\u7684\u4ea4\u70b9\uff0c\u5373y=x=x²\uff0c\u89e3\u65b9\u7a0b\u5f97\u4e24\u4ea4\u70b9\u4e3a\uff080\uff0c0\uff09\u548c\uff081\uff0c1\uff09
\u2234 S2=\u222b\uff080\uff0c1\uff09\uff08x-x²\uff09dx=½x²-1/3x³\u4e28\uff080\uff0c1\uff09=\uff08½-1/3\uff09-0=1/6
\u7531\u4e8e\u4e09\u6761\u7ebf\u5747\u4ea4\u4e8e\u539f\u70b9\uff080\uff0c0\uff09
\u2234 \u6240\u6c42\u9762\u79efS=S1-S2=4/3-1/6=7/6
\u5206\u6790\uff1a\u6240\u6c42\u9762\u79ef\uff1d\u629b\u7269\u7ebf\u4e0e\u76f4\u7ebfy\uff1d2x\u56f4\u6210\u7684\u9762\u79ef\uff0d\u629b\u7269\u7ebf\u4e0e\u76f4\u7ebfy\uff1dx\u56f4\u6210\u7684\u9762\u79ef\uff01
\u89e3\uff1a
\u8054\u7acb\uff5by\uff1dx²\uff0c \u5f97 \uff5bx1\uff1d0
\uff5by\uff1d2x\uff0c \uff5bx2\uff1d2
\u2234\u629b\u7269\u7ebf\u4e0e\u76f4\u7ebfy\uff1d2x\u6240\u56f4\u6210\u7684\u9762\u79ef\u4e3a\uff1a
S1\uff1d\u222b(0,2)\uff082x\uff0dx²\uff09dx
\uff1d[x²\uff0d(1/3)x³]|(0,2)
\uff1d(4\uff0d8/3)\uff0d0
\uff1d4/3.
\u8054\u7acb\uff5by\uff1dx²\uff0c \u5f97 \uff5bx1\uff1d0
\uff5by\uff1dx\uff0c \uff5bx2\uff1d1
\u2234\u629b\u7269\u7ebf\u4e0e\u76f4\u7ebfy\uff1dx\u6240\u56f4\u6210\u7684\u9762\u79ef\u4e3a\uff1a
S2\uff1d\u222b(0,1)\uff08x\uff0dx²\uff09dx
\uff1d[(1/2)x²\uff0d(1/3)x³]|(0,1)
\uff1d1/2\uff0d1/3
\uff1d1/6.
\u2234S\uff1dS1\uff0dS2\uff1d4/3\uff0d1/6\uff1d7/6.
取距离y轴为x的宽度为dx的一个微元小窄条，其微元面积dS应为分段函数，分为[0,1]和(1,2]两个区间进行表达。
于是围成图形的面积为
S=∫dS=∫ (0,1) (2x-x)dx +∫ (1,2) (2x-x^2)dx
=(1/2*x^2) | (0,1) +(x^2-1/3*x^3) | (1,2)
=1/2+3-7/3=7/6
----------------
满意请采纳哟^ ^~
可以用定积分计算。
当x²=2x时，X=2或0
S=∫02（2x）dx - ∫02（x²）dx=2x丨02dx - x²丨02dx =2X2+2²-（2X0+0²）=8
【0为下标，2为上标】
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