Given x=4e^t cos(t), y=4e^t sin(t) t is greater than or equal to 0 to and less t

Question

Given x=4e^t cos(t), y=4e^t sin(t) t is greater than or equal to 0 to and less than or equal to pi (t is front 0 to pi). Find the exact length of the curve. Given x=4e^t cos(t), y=4e^t sin(t) t is greater than or equal to 0 to and less than or equal to pi (t is front 0 to pi). Find the exact length of the curve. Given x=4e^t cos(t), y=4e^t sin(t) t is greater than or equal to 0 to and less than or equal to pi (t is front 0 to pi). Find the exact length of the curve. Given x=4e^t cos(t), y=4e^t sin(t) t is greater than or equal to 0 to and less than or equal to pi (t is front 0 to pi). Find the exact length of the curve.

Explanation / Answer

given x=4e^t cos(t), y=4e^t sin(t)

differentiate with respect to t

dx/dt=4e^t cos(t)-4e^t sin(t), dy/dt=4e^t sin(t)+4e^t cos(t)

ds=[(dx/dt)2+(dy/dt)2] dt

ds=[(4e^t cos(t)-4e^t sin(t))2+(4e^t sin(t)+4e^t cos(t))2] dt

ds=4e^t[( cos(t)- sin(t))2+(sin(t)+ cos(t))2] dt

ds=4e^t[( cos2(t)+ sin2(t)-2cos(t)sin(t))+(sin2(t)+ cos2(t)+2sin(t)cos(t))] dt

ds=4e^t[(1-2cos(t)sin(t))+(1+2sin(t)cos(t))] dt

ds=4e^t2 dt

ds=(42)e^t dt

length of curve ,S=[0 to ]ds

length of curve ,S=[0 to ](42)e^t dt

length of curve ,S=[0 to ](42)e^t

length of curve ,S=(42)(e -e0)

length of curve ,S=(42)(e -1)

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