Explanation Given info: f ( x , y , z ) = e z Integration over the tetrahedron W as show in figure. Concept used: Frist find the equation of plane using following Formula a ( x − x 0 ) + b ( y − y 0 ) + c ( z − z 0 ) = 0 Where ( a , b , c ) is normal vector to the plane? ( x 0 , y 0 , z 0 ) is any point on the plane? With the help of this equation find the lower and upper limit. Take projection on x y plane of the given region. After that find out the limit of x and y . Calculation: Finding the two vectors in the plane P R → = ( 0 − 4 ) i ∧ + ( 0 − 0 ) j ∧ + ( 6 − 0 ) k ∧ = − 4 i ∧ + 0 j ∧ + 6 k ∧ P Q → = ( 0 − 4 ) i ∧ + ( 4 − 0 ) j ∧ + ( 0 − 0 ) k ∧ = − 4 i ∧ + 4 j ∧ + 0 k ∧ Normal vector on the plane is given by n → = P R → × P Q → = ( − 4 i ∧ + 0 j ∧ + 6 k ∧ ) × ( − 4 i ∧ + 4 j ∧ + 0 k ∧ ) = { − 4 i ∧ × ( − 4 i ∧ + 4 j ∧ ) + { 0 j ∧ × ( − 4 i ∧ + 4 j ∧ + 0 k ∧ ) } + 6 k ∧ + × { − 4 i ∧ + 4 j ∧ + 0 k ∧ ) = + 16 i ∧ × i ∧ − 16 i ∧ × i ∧ + 0 → − 24 k ∧ × j ∧ + 24 k ∧ × j ∧ + 0 → ( i ∧ × i ∧ = 0 , i ∧ × i ∧ = k ∧ , k ∧ × i ∧ = j ∧ , k ∧ × j ∧ = − i ∧ ) = − 16 k ∧ − 24 j ∧ − 24 i ∧ = − 24 i ∧ − 24 j ∧ − 16 k ∧ Hence normal vector ( − 24 , − 24 , − 16 ) Equation of plane a ( x − x 0 ) + b ( y − y 0 ) + ( Z − Z 0 ) = 0 ( a , b , c ) = ( − 24 , − 24 , − 16 ) ( x 0 , y 0 , z 0 ) ≡ ( 4 , 0 , 0 ) Putting these value to find out the equation of plane. − 24 ( x − 4 ) − 24 ( y − 0 ) − 16 ( z − 0 ) = 0 3 ( x − 4 ) + 3 y + 2 z = 0 3 x + 3 y + 2 z = 12 x 4 + y 4 + z 6 = 1 ( This equation of plane can be obtained directly by intercept formula x a + y b + z c = 1 ) Where a , b , c are x , y and z intercept respectively z 6 = 1 − x 4 − y 4 z = 6 ( 1 − x 4 − y 4 ) This gives upper limit of z , and z ≥ 0 0 ≤ z ≤ 6 ( 1 − x 4 − y 4 ) To decide the limit of x and y take projection on x y plane of this region. To find the equation of inclined line take projection of x 4 + y 4 + z 6 = 1 on x y plane put z = 0 x + y = 4 Upper limit of y is restricted by x + y = 4 y = 4 − x And lower limit of y is 0 0 ≤ y ≤ 4 − x 0 ≤ x ≤ 4 ( from figure ) W: 0 ≤ x ≤ 4 , 0 ≤ y ≤ 4 − x , 0 ≤ z ≤ 6 ( 1 − x 4 − y 4 ) Now you are able to integrate ∫ ∫ W ∫ f ( x , y , z ) d v = ∫ ∫ W ∫ e z ...

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.3, Problem 16E

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Calculate the integral of $f (x, y, z) = z$ over the tetrahedron $w$ in figure

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Chapter 16.3, Problem 15E

Chapter 16.3, Problem 17E

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Chapter 16 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

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Calculate the integral of f ( x , y , z ) = z over the tetrahedron w in figure

Solution Summary: The author explains how to determine the integral of f (x, y ) = e z over the tetrahedron w.

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Calculate the integral of $f (x, y, z) = z$ over the tetrahedron $w$ in figure

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Chapter 16 Solutions

Calculate the integral of f ( x , y , z ) = z over the tetrahedron w in figure

Solution Summary: The author explains how to determine the integral of f (x, y ) = e z over the tetrahedron w.

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Calculate the integral of f(x,y,z)=z over the tetrahedron win figure

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Chapter 16 Solutions

Calculate the integral of $f (x, y, z) = z$ over the tetrahedron $w$ in figure