   Chapter 10, Problem 28RE

Chapter
Section
Textbook Problem

# Find the area enclosed by the loop of the curve in Exercise 27.

To determine

To find: The area of the region which is enclosed inside the loop of the curve x=t33t and y=t2+t+1 .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=t33t (1)

The parametric equation for the variable y is as follows.

y=t2+t+1 (2)

Calculation:

The limits of the formation of the curve are found to be 4t3 .

The curve can be magnified to know the exact point of intersection of the curve by restricting the limits as 2.4t1.6 .

Summarize the values of x and y using the limit 2.4t1.6 as shown in table (1) below.

 t x y -2.4 -6.624 4.36 -2.2 -4.048 3.64 -2 -2 3 -1.8 -0.432 2.44 -1.6 0.704 1.96 -1.4 1.456 1.56 -1.2 1.872 1.24 -1 2 1 -0.8 1.888 0.84 -0.6 1.584 0.76 -0.4 1.136 0.76 -0.2 0.592 0.84 0 0 1 0.2 -0.592 1.24 0.4 -1.136 1.56 0.6 -1.584 1.96 0.8 -1.888 2.44 1 -2 3 1.2 -1.872 3.64 1.4 -1.456 4.36 1.6 -0.704 5.16 1.8 0.432 6.04 2 2 7 2.2 4.048 8.04 2.4 6.624 9.16

Draw the graph using the table (1) as shown in the figure (1) below.

Refer to the graph, the point of intersection of the curve is (2,3) .

Refer to the table, to get this point (2,3) at t=2 and t=1 ( 2t1 ).

Calculate the area enclosed inside the loop by using the formula.

A=abydx (3)

Differentiate equation (2) with respect to t .

dxdt=3t2t

dx=(3t2t)dt (4)

Substitute the expressions from equations (4) and (1) in equation (3) as below.

A=21(t2+t+1)(3t23)dt

Integrate with respect to t and apply limits from 2 to 1

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