   Chapter 10.3, Problem 82E

Chapter
Section
Textbook Problem

Area of Simple Closed Curves In Exercises 81-86, use a computer algebra system and the result of Exercises 77 to match the closed curve with its area. (These exercises were based on “The Surveyor’s Area Formula” by Bart Braden, College Mathematics Journal , September 1986, pp335-337, 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 Astroid :     ( 0 ≤ t ≤ 2 π ) x = a cos 3 t y = a sin 3 t To determine

To calculate: The close curve area of an astroid, x=acos3t and y=asin3t in the interval 0t2π by the graphing utility and match the result with the given areas.

Explanation

Given:

Parametric equations are: x=acos3t,y=bsin3t and the interval, 0t2π.

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 asteroid, x=acos3t,y=asin3t.

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 asteroid,

So, differentiate f(t) with respect to t

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