Explanation Given: The inequality: ( x 2 + y 2 + z 2 + 8 ) 2 ≤ 36 ( x 2 + y 2 ) Formula used: The volume of the torus is: V = ∫ 0 2 π ∫ r 1 r 2 ∫ z 1 z 2 r d z d r d θ The integration formula: ∫ a 2 − x 2 d x = x ( a 2 − x 2 ) 2 + a 2 2 sin − 1 ( x a ) + C Calculation: Consider the provided inequality, ( x 2 + y 2 + z 2 + 8 ) 2 ≤ 36 ( x 2 + y 2 ) Convert the inequality for the points ( x , y , z ) in the cylindrical form that is for the point ( r cos θ , r sin θ , z ) From this, ( ( r cos θ ) 2 + ( r cos θ ) 2 + z 2 + 8 ) 2 ≤ 36 ( ( r cos θ ) 2 + ( r sin θ ) 2 ) ( r 2 + z 2 + 8 ) 2 ≤ 36 r 2 Take square root both the sides to get, r 2 + z 2 + 8 ≤ 6 r Form this, r 2 − 6 r + z 2 + 8 ≤ 0 r 2 − 2 ( 3 ) r + 9 − 9 + z 2 + 8 ≤ 0 ( r − 3 ) 2 + z 2 ≤ 1 …… (1) Draw the above equation in the r-z plane. The graph obtained is shown below. Thus, the equation (1) represents a torus about z -axis. Therefore, the volume of the torus is: V = ∫ 0 2 π ∫ r 1 r 2 ∫ z 1 z 2 r d z d r d θ The limits are as follows. − 1 − ( r − 3 ) 2 ≤ z ≤ 1 − ( r − 3 ) 2 2 ≤ r ≤ 4 0 ≤ θ ≤ 2 π Put the limits in the formula to get, V = ∫ 0 2 π ∫ 2 4 ∫ − 1 − ( r − 3 ) 2 1 − ( r − 3 ) 2 r d z d r d θ Solve the iterated integral to get the volume of the torus, V = 2 ∫ 0 2 π ∫ 2 4 ∫ 0 1 − ( r − 3 ) 2 r d z d r d θ = 2 ∫ 0 2 π ∫ 2 4 ( ∫ 0 1 − ( r − 3 ) 2 r d z ) d r d θ = 2 ∫ 0 2 π ∫ 2 4 ( r 1 − ( r − 3 ) 2 ) d r d θ = 2 ∫ 0 2 π ∫ 2 4 ( r − r 2 + 6 r − 9 ) d r d θ On further simplification, V = 2 ∫ 0 2 π ∫ 2 4 ( ( − 1 2 ( − 2 r + 6 ) + 3 ) − r 2 + 6 r − 8 ) d r d θ = 2 ∫ 0 2 π ∫ 2 4 ( − 1 2 ( − 2 r + 6 ) − r 2 + 6 r − 8 + 3 − r 2 + 6 r − 8 </

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Publisher: Cengage Learning

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Chapter 14.7, Problem 47E

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To calculate: The volume of the region of points (x,y,z) such that (x2+y2+z2+8)2≤36(x2+y2).

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Chapter 14.7, Problem 46E

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Chapter 14 Solutions

Multivariable Calculus

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The volume of the region of points ( x , y , z ) such that ( x 2 + y 2 + z 2 + 8 ) 2 ≤ 36 ( x 2 + y 2 ) .

Solution Summary: The author explains how to calculate the volume of the region of points (x,y,z).

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To calculate: The volume of the region of points (x,y,z) such that (x2+y2+z2+8)2≤36(x2+y2).

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Chapter 14 Solutions