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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the area of the surface obtained by rotating the given curve about the x-axis.

41. Chapter 10, Problem 41RE, Find the area of the surface obtained by rotating the given curve about the x-axis. 41. , , 1  t  4 , example  1, Chapter 10, Problem 41RE, Find the area of the surface obtained by rotating the given curve about the x-axis. 41. , , 1  t  4 , example  2, 1 ≤ t ≤ 4

To determine

To find: The area of the surface obtaining by rotating the curve x=4t and y=t33+12t2 for 1t4 about x-axis.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=4t (1)

The parametric equation for the variable y is as follows.

y=t33+12t2 (2)

Calculation:

Differentiate the equation (1) with respect to t .

dxdt=4×12(t12)

dxdt=2(t12)

dxdt=2t (3)

Differentiate the equation (2) with respect to t .

dydt=3t2t3

dydt=t21t3 (4)

Surface area is determined for the limits of 1 to 4 .

Calculate the surface area the curve rotating about x axis using the formula.

S=142πy(dxdt)2+(dydt)2dt (5)

Substitute the expressions from equation (2), (3) and (4) into equation (5).

S=142πy(dxdt)2+(dydt)2dt=142π(t33+12t2)(2t)2+(t21t3)2dt=142π(t33+12t2)4t+t4+1t6(2t2×1t3)dt

S=142π(t33+12t2)4t+t

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Chapter 10 Solutions

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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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