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) $697.43 \mathrm {~cm} ^ { 2 }$

Question

Quizplus · Accepted Answer

The lateral surface area of the component is calculated using the formula for the lateral surface area of a surface of revolution. The integral for the lateral surface area is $\int_{0}^{2\pi} \sqrt{(dz/dt)^2 + (dx/dt)^2 + (dy/dt)^2} \, dt$. After evaluating the integral, the lateral surface area is approximately $716.28 \mathrm {~cm} ^ { 2 }$.

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

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