Homework Soursex3tt3 y3t3 0
x=3t-t^3, y=3t^3, 0<_t<_1 about the x axis
Solution
sqrt [(dx/dt)^2 + (dy/dt)^2]
= sqrt [(3 - 3t^2)^2 + (6t)^2]
= sqrt [(9 - 18t^2 + 9t^4) + 36t^2]
= sqrt (9 + 18t^2 + 9t^4)
= sqrt(3 + 3t^2)^2
= 3 + 3t^2.
So, the surface area equals
2 y sqrt [(dx/dt)^2 + (dy/dt)^2] dt
= (t = 0 to 1) 2 * 3t^2 * (3 + 3t^2) dt
= (t = 0 to 1) 18 (t^2 + t^4) dt
= 18 (t^3/3 + t^5/5) {for t = 0 to 1}
= 48/5.
![x=3t-t^3, y=3t^3, 0<_t<_1 about the x axisSolutionsqrt [(dx/dt)^2 + (dy/dt)^2] = sqrt [(3 - 3t^2)^2 + (6t)^2] = sqrt [(9 - 18t^2 + 9t^4) + 36t^2] = sqrt (](/WebImages/37/x3tt3-y3t3-0-1112929-1761590517-0.webp "x=3t-t^3, y=3t^3, 0<_t<_1 about the x axisSolutionsqrt [(dx/dt)^2 + (dy/dt)^2] = sqrt [(3 - 3t^2)^2 + (6t)^2] = sqrt [(9 - 18t^2 + 9t^4) + 36t^2] = sqrt (")
