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A=∫y(t)dxdtdtA = \int y(t) \frac{dx}{dt} dtA=∫y(t)dtdxdt, integrating with respect to ttt.
Use dx/dtdx/dtdx/dt and multiply by dtdtdt to find dxdxdx.
dx/dt=1dx/dt = 1dx/dt=1.
From t=1t=1t=1 to t=5t=5t=5.
I = ∫15(1−1/t)dt\int_{1}^{5} (1 - 1/t) dt∫15(1−1/t)dt.
I=4−ln(5)I = 4 - \ln(5)I=4−ln(5) from the integration process.
x=3tx = 3tx=3t, y=sin(t)y = \sin(t)y=sin(t).
t=0,π,t = 0, \pi,t=0,π, and 2π2\pi2π.
dx=3dtdx = 3dtdx=3dt.
I=6I = 6I=6.
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