A Tractor Engine Has a Steel Component with a Circular $\mathbf { r } ( t ) = 6 \cos t \mathbf { i } + 6 \sin t \mathbf { j }$

Multiple Choice

A tractor engine has a steel component with a circular base modeled by the vector-valued function $\mathbf { r } ( t ) = 6 \cos t \mathbf { i } + 6 \sin t \mathbf { j }$ . Its height is given by $z = 1 + y ^ { 2 }$ . (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places.

A) $753.98 \mathrm {~cm} ^ { 2 }$
B) $735.13 \mathrm {~cm} ^ { 2 }$
C) $716.28 \mathrm {~cm} ^ { 2 }$
D) $245.04 \mathrm {~cm} ^ { 2 }$
E) $697.43 \mathrm {~cm} ^ { 2 }$