   Chapter 10.3, Problem 86E

Chapter
Section
Textbook Problem

# Areas of Simple Closed Curves In Exercises 81-86, use a computer algebra system and the result of Exercise 77 to match the closed curse with its area. (These exercises were based on “The Surveyor’s Area Formula'’ by Bart Braden, College Mathematics Journal, September 1986, pp. 335-337, by permission of the author.)(a) 8 3 a b (b) 3 8 π a 2 (c) 2 π a 2 (d) π a b (e) 2 π a b (f) 6 π a 2 Teardrop: ( 0 ≤ t ≤ 2 π ) x = 2 a cos t − a sin 2 t y = b sin t To determine

To calculate: The close curve area of a teardrop, x=2acostasin2t and y=bsint in the interval 0t2π with graphing utility and match the result with the given areas.

Explanation

Given:

Parametric equations are, x=2acostasin2t,y=bsint and the figure:

Formula used:

If y is a continuous function of x on the interval axb, where x=f(t) and y=g(t), then,

|abydx|=|t1t2g(t)f(t)dt| where f(t1)=a and f(t2)=b and both g and f are continuous on [t1,t2].

Calculation:

Consider the equation of the teardrop, x=2acostasin2t,y=bsint.

Assume that f(t)=x and g(t)=y.

Now use the formula, |abydx|=|t1t2g(t)f(t)dt| to find the area of the teardrop,

So, differentiate f(t) with respect to t

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