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Complete Example 2 by showing that

-π/4π/4k38cos3θ-sec3θdθ=k9(172+3ln(2-1))

Short Answer

Expert verified

The mass of semicircular lamina is

m=k9(172+3ln(2-1))

Step by step solution

01

Given information

The expression is

-π/4π/4k38cos3θ-sec3θdθ=k9(172+3ln(2-1))
02

Calculation

Density ρ(r)=kr

Equation of circle

r=2cosθ

Equation of line x=1in polar form r=secθ

Mass of the plate

m=Ωρ(x,y)dAm=-π/4π/4secθ2cosθkr2drdθ

Integrate the inner integral with respect to rfirst.

m=k-π/4π/4r33secθ2cosθdθm=k-n/4π/48cos3θ-sec3θ3dθ

Substitute the limits

m=k3-π/4π/48cos3θ-sec3θdθ

m=k3812{9sinθ+sin3θ}-12tanθsecθ-lncosθ2-sinθ2+lncosθ2+sinθ2-π/4π/4

m=k38129sinπ4+sin3π4-12tanπ4secπ4-lncosπ8-sinπ8+lncosπ8+sinπ8d-8129sin-π4+sin-3π4+12tan-π4sec-π4-lncos-π8lncos-π8+sin-π8+3ln-π8

m=k9(172+3ln(2-1))

Thus, the mass of semicircular lamina is

m=k9(172+3ln(2-1))

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