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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.

42. x = 2 + 3t, y = cosh 3t, 0 ≤ t ≤ 1

To determine

To find: The area of the surface obtaining by rotating the curve x=2+3t and y=cosh3t for 0t1 about x-axis.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=2+3t (1)

The parametric equation for the variable y is as follows.

y=cosh3t (2)

Calculation:

Differentiate equation (1) with respect to t .

dxdt=3 (3)

Differentiate equation (2) with respect to t .

dydt=3sinh3t (4)

Surface area is determined for the limits of 0 to 1 .

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

S=012πy(dxdt)2+(dydt)2dt (5)

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

S=012πy(dxdt)2+(dydt)2dt=012π(cosh3t)(3)2+(3sinh3t)2dt=012π(cosh3t)9+9sinh23tdt

S=012π(cosh3t)9(1+sinh23t)d

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