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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 enclosed by the curve x = t2 − 2t, y = t and the y-axis.

To determine

To find: The area enclosed by the x axis and the curve for the parametric equation x=t2+2t and y=t .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=t2+2t (1)

The parametric equation for the variable y is as follows.

y=ty2=t

y2=t (2)

Substitute (y2) for t in equation (1).

x=t2+2t=(y2)2+2y2

x=y4+2y2 (3)

Substitute 0 for t in equation (1).

x=t2+2t=(0)+2(0)=0

Substitute 0 for t in equation (2).

y2=ty=0

Substitute 2 for t in equation (1).

x=t2+2t=(2)2+2(2)=4+4=8

Substitute 2 for t in equation (2).

y2=t=(2)=2

The values of x and y for each step value of t is tabulated in the below table.

t 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 0 0.36 0.64 0.84 0.96 1 0.96 0.84 0.64 0.36 0
y 0 0.44 0.63 0.77 0.89 1 1.09 1.18 1.26 1.34 1.41

Graph:

The graph is plotted for the values of x and y is shown below in figure 1

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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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