Respuesta :
Answer:
Arc length [tex]=\int_0^{\pi} \sqrt{1+[(4.5sin(4.5x))]^2}\ dx[/tex]
Arc length [tex]=9.75053[/tex]
Step-by-step explanation:
The arc length of the curve is given by [tex]\int_a^b \sqrt{1+[f'(x)]^2}\ dx[/tex]
Here, [tex]f(x)=\int_0^{4.5x}sin(t) \ dt[/tex] interval [tex][0, \pi][/tex]
Now, [tex]f'(x)=\frac{\mathrm{d} }{\mathrm{d} x} \int_0^{4.5x}sin(t) \ dt[/tex]
[tex]f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}\left ( [-cos(t)]_0^{4.5x} \right )[/tex]
[tex]f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}\left ( -cos(4.5x)+1 \right )[/tex]
[tex]f'(x)=4.5sin(4.5x)[/tex]
Now, the arc length is [tex]\int_0^{\pi} \sqrt{1+[f'(x)]^2}\ dx[/tex]
[tex]\int_0^{\pi} \sqrt{1+[(4.5sin(4.5x))]^2}\ dx[/tex]
After solving, Arc length [tex]=9.75053[/tex]
