y=0.5x^2 x^2+y^2=8 两条曲线围成的图形的面积两部分都要算 用定积分的应用
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y=0.5x^2 x^2+y^2=8 两条曲线围成的图形的面积两部分都要算 用定积分的应用
y=0.5x^2 x^2+y^2=8 两条曲线围成的图形的面积
两部分都要算 用定积分的应用
y=0.5x^2 x^2+y^2=8 两条曲线围成的图形的面积两部分都要算 用定积分的应用
代入得
x^2+0.25x^4=8
x^4+4x^2-32=0
(x^2-4)(x^2+8)=0
x^2=4
x=2,x=-2
y=2
所以交点(2,2),(-2,2)
围成的图形在第一和第二象限,且关于y轴对称
所以只需计算第一象限内的
此时x^2+y^2=8则y=√(8-x^2)
所以S/2=∫(0到2)[√(8-x^2)-0.5x^2]dx
先算∫(0到2)√(8-x^2)dx
令x=2√2cosa
8-x^2=8(sina)^2
√(8-x^2)=2√2sina
则dx=-2√2sina
x=0,a=π/2,
x=2,a=π/4
所以∫(0到2)√(8-x^2)dx
=∫(π/2到π/4)√(8-x^2)dx
=∫(π/2到π/4)2√2sina*(-2√2sina)da
=-8∫(π/2到π/4)(sina)^2da
=-4∫(π/2到π/4)(1-cos2a)/2d2a
=-2∫(π/2到π/4)(1-cos2a)d2a
=-2(2a-sin2a)(π/2到π/4)
=π+2
∫(0到2)0.5x^2dx
=(0.5/3)x^3(0到2)
=4/3
所以S/2=π+2-4/3=π+2/3
所以S=2π+4/3
只有一部分吧,先解这个方程组,,画出图形,再确定积分的上下限
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