## [數學] 請幫我

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### [數學] 請幫我

2.x^2/a^2+y^2/b^2=1

y=b/a √(a^2- x^2)

∫b/a √(a^2- x^2)dx ,at(-a, a)= b/a∫√(a^2- x^2)dx ,at(-a, a)

= b/a a^2[1/2*arcsin(x/a)+1/2*x/a^2*√(a^2- x^2)] ,at(-a, a)
= ab/2[arcsin(x/a)+x/a^2*√(a^2- x^2)] ,at(-a, a)
= abπ/2

6.y=9-x^2 ,at(0, 3)

=2π∫x√(1+4x^2) dx=4π∫x√(1/4+x^2) dx

u=arctan(2x)
∫x√(1+4x^2) dx=∫1/2tanu*1/2 secu*1/2 sec^2u du
=1/8∫(tanu*sec^3u) du
=1/8∫(sinu/cos^4u) du
=-1/8∫1/cos^4u dcosu
=3/8*1/cos^3u

u=arctan(2x) ,x at(0, 3) ,u at(0, arctan6) ,cosu at(1,1/√37)

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3.1 繞x軸旋轉體積=π∫f^2(x) dx

y=x√(4-x^2)
f^2(x)= x^2(4-x^2)=4 x^2-x^4
π∫(4 x^2-x^4) dx=π(4/3 x^3-1/5 x^5) ,x at(-2,2)
=128π/15

3.2
π∫(-1/2 x+4) ^2- √x^2 dx , x at(0,8)
=π∫(1/4 x^2-4x+16) -x dx
=π∫(1/4 x^2-5x+16) dx
=π(1/12 x^3-5/2x^2+16x) , x at(0,8)
=32π/3

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4.1
π∫f^2(x) dx=π∫100/x^4 dx=-π100/3 x^3 , x at(1,5)
=164π/5

4.2繞y軸旋轉體積=2π∫xf(x) dx
2π∫xf(x) dx=2π∫10/x dx=20πlnx x at(1,5)
=20π*ln5

4.3 line y=10
Let y1=y-10
y1=10/x^2-10
π∫f^2(x) dx=π∫100/x^4-200/ x^2+100 dx
=π(-100/3x^3+200/x+100x) ,x at(1,5)
=4096π/15

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4.3

=1904π/15

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pi, 圓周率

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