Explanation Given: x 2 + y 2 + z 2 = 36 and ( x − 3 ) 2 + y 2 = 9 . Formula used: By triple integration of cylindrical coordinates, volume of solid can be defined as V = ∭ Q f ( x , y , z ) d V = ∭ Q f ( r cos θ , r sin θ , z ) r d z d r d θ Rectangular conversion equations of cylindrical coordinates are, x = r cos θ y = r sin θ z = z Relation between x , y , and r is given as, x 2 + y 2 = r 2 As, sin 2 θ + cos 2 θ = 1 Trigonometry formula, sin 3 θ = 3 sin − 1 θ − 4 sin 3 θ Calculation: Consider the figure of provided solid, Consider the available equation, x 2 + y 2 + z 2 = 36 , and solve it as, x 2 + y 2 + z 2 = 36 r 2 + z 2 = 36 z 2 = 36 − r 2 z = ± 36 − r 2 Limits on z are, 0 ≤ z ≤ 36 − r 2 For the other availableequation, ( x − 3 ) 2 + y 2 = 9 x 2 + 9 − 6 x + y 2 = 9 x 2 + y 2 = 6 x r 2 = 6 r cos θ Or, r = 6 cos θ Consider R be the surface projection on r θ plane. The limits on r are 0 ≤ r ≤ 6 cos θ and on θ is 0 ≤ θ ≤ π . Volume of solid is given as, V = ∭ Q f ( x , y , z ) d V = ∫ 0 π ∫ 0 6 cos θ ∫ − 36 − r 2 36 − r 2 r d z d r d θ = ∫ 0 π ∫ 0 6 cos θ r [ z ] − 36 − r 2 36 − r 2 d r d θ Further, ∫ 0 π ∫ 0 6 cos θ ∫ − 36 − r 2 36 − r 2 r d z d r d θ = ∫ 0 π ∫ 0 6 cos θ r [ 36 − r 2 − ( − 36 − r 2 ) ] d r d θ = 2 ∫ 0 π ∫ 0 6 cos θ r 36 − r 2 d r d θ Set 36 − r 2 = t and on differentiating both the sides with respect to “ r” , we get d d r ( 36 − r 2 ) = d d r ( t ) − 2 r = d t d r − 2 r d r = d t As, “ r” changes from 0 to 6 cos θ , “ t” changes from 36 to 36 sin 2 θ

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Chapter 14.7, Problem 15E

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To calculate: By means of Cylindrical coordinates find out the inside volume of solid for x2+y2+z2=16 and (x−3)2+y2=9.

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Chapter 14.7, Problem 14E

Chapter 14.7, Problem 16E

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Chapter 14 Solutions

CALCULUS LL UPGRADE CUSTOM

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By means of Cylindrical coordinates find out the inside volume of solid for x 2 + y 2 + z 2 = 16 and ( x − 3 ) 2 + y 2 = 9 .

Solution Summary: The author explains how to calculate the inside volume of solid by triple integration of cylindrical coordinates.

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To calculate: By means of Cylindrical coordinates find out the inside volume of solid for x2+y2+z2=16 and (x−3)2+y2=9.

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Chapter 14 Solutions