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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 length of the curve.

37. x = 3t2, y = 2t3, 0 ≤ t ≤ 2

To determine

To find: the length of the curve x=3t2 and y=2t3 for 0t2 .

Explanation

Given:

The Parametric equation for the variable x is as follows.

x=3t2 (1)

The Parametric equation for the variable y is as follows.

y=2t3 (2)

Calculation:

Differentiate equation (1) with respect to t we get,

dxdt=6t (3)

Differentiate equation (2) with respect to t we get,

dydt=6t2 (4)

Length is determined for the limits of 0 to 2 .

Calculate the length of the curve using the formula.

L=02(dxdt)2+(dydt)2dt (5)

Substitute the expressions from equation (3) and (4) in equation (5).

L=02(dxdt)2+(dydt)2dt=02(6t)2+(6t2)2dt=02(36t2+36t4)dt

=0236t2(1+t2)dt

=026t(1+t2)dt (6)

To make the integration simple take

u=(1+t2) (7)

When the value of t=0 , the value of u=1

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