   Chapter 10.2, Problem 32E

Chapter
Section
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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