Explanation The Cartesian equation of the curve is given as, x = t 3 − 3 t and y = t 2 + t + 1 . From Exercise 27, the graph of the curve with Cartesian equation x = t 3 − 3 t and y = t 2 + t + 1 is shown below in Figure 1. From Figure 1, it is observed that the loop is contained between the points ( − 2 , 3 ) and ( 2 , 1 ) . For t = − 2 the values of x and y are, x = ( − 2 ) 3 − 3 ( − 2 ) = − 8 + 6 = − 2 y = ( − 2 ) 2 + ( − 2 ) + 1 = 4 − 2 + 1 = 3 For t = 0 the values of x and y are, x = ( 0 ) 3 − 3 ( 0 ) = 0 y = ( 0 ) 2 + ( 0 ) + 1 = 1 For t = 1 the values of x and y are, x = t 3 − 3 t = 1 3 − 3 ( 1 ) = − 2 y = t 2 + t + 1 = 1 2 + 1 + 1 = 3 That is, when − 2 ≤ t ≤ 0 , the curve follows the path from the point ( − 2 , 3 ) to ( 2 , 1 ) and when 0 ≤ t ≤ 1 , the curve follows the path from the ( 2 , 1 ) to ( − 2 , 3 ) . That is, the loop is obtained for the t values between − 2 ≤ t ≤ 1 . The area under a curve y = F ( x ) from a to b is given by, A = ∫ a b y d x Differentiate x = t 3 − 3 t with respect to t . d x d t = 3 t 2 − 3 d x = ( 3 t 2 − 3 ) d t Substitute y = t 2 + t + 1 , a = − 2 , b = 1 and d x = ( 3 t 2 − 3 ) d t in A = ∫ a b y d x

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ISBN: 9781133425946

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Chapter 9, Problem 28RE

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The area of the region enclosed by the loop of the curve x=t3−3t and y=t2+t+1.

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Chapter 9, Problem 27RE

Chapter 9, Problem 29RE

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Bundle: Stewart, Essential Calculus: Early Transcendentals, 2nd (hardound) + WebAssign Printed Access Card for Stewart's Essential Calculus: Early ... Multi-Term + WebAssign - Start Smart Guide

Ch. 9.1 - 14 Sketch the curve by using the parametric...Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - (a) Eliminate the parameter to find a Cartesian...Ch. 9.1 - (a) Eliminate the parameter to find a Cartesian...

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Prob. 5RE Ch. 9 - Prob. 6RE Ch. 9 - Prob. 7RE Ch. 9 - Sketch the region consisting of points whose polar...Ch. 9 - Prob. 9RE Ch. 9 - Prob. 10RE Ch. 9 - Sketch the polar curve. 11. r = cos 3 Ch. 9 - Sketch the polar curve. 12. r = 3 + cos 3 Ch. 9 - Sketch the polar curve. 13. r = 1 + cos 2 Ch. 9 - Sketch the polar curve. 14. r = 2 cos(/2)Ch. 9 - Sketch the polar curve. 15. r=31+2sin Ch. 9 - Sketch the polar curve. 16. r=322cos Ch. 9 - Find a polar equation for the curve represented by...Ch. 9 - Find a polar equation for the curve represented by...Ch. 9 - The curve with polar equation r = (sin )/ is...Ch. 9 - Prob. 20RE Ch. 9 - Find the slope of the tangent line to the given...Ch. 9 - Find the slope of the tangent line to the given...Ch. 9 - Find the slope of the tangent line to the given...Ch. 9 - Find the slope of the tangent line to the given...Ch. 9 - Find dy/dx and d2y/dx2. 25. x = t + sin t, y = t ...Ch. 9 - Prob. 26RE Ch. 9 - Prob. 27RE Ch. 9 - Prob. 28RE Ch. 9 - At what points does the curve x = 2a cos t a cos...Ch. 9 - 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The area of the region enclosed by the loop of the curve x = t 3 − 3 t and y = t 2 + t + 1 .

Solution Summary: The graph of the curve with the Cartesian equation x=t3-3t and

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The area of the region enclosed by the loop of the curve x=t3−3t and y=t2+t+1.

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Chapter 9 Solutions