Differential Elements
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DIFFERENTIAL ELEMENTS
a.) Illustration
b.) Derivation of formula The volume of cylindrical element is... 𝒅𝑽 = 𝝅𝒙𝟐 𝒅𝒚 The sum of the cylindrical elements from 0 to r is a hemisphere, twice the hemisphere will give the volume of the sphere. Thus, 𝒓 𝐕 = 𝟐𝛑 ∫𝟎 𝒙𝟐 𝒅𝒚 From the equation of the circle 𝒙𝟐 + 𝒚𝟐 = 𝒓𝟐 ; 𝒙𝟐 = 𝒓𝟐 − 𝒚𝟐 . 𝒓 𝑽 = 𝟐𝝅 ∫𝟎 (𝒓𝟐 − 𝒚𝟐 )𝒅𝒚 𝑽 = 𝟐𝝅 [𝒓𝟐 𝒚 −
𝒚𝟑
𝟑 𝒓𝟑
]
𝑽 = 𝟐𝝅 [(𝒓𝟑 − 𝟑 ) − (𝟎 − 𝑽 = 𝟐𝝅 [ 𝑽=
𝟒𝝅𝒓𝟑 𝟑
𝟐𝒓𝟑 𝟑
𝟎𝟑 𝟑
)]
]
.
c.) Sample Problem for Differential Area
Problem: Calculate the surface area of a sphere with radius 3.2 cm Solution: Surface area of sphere = 4π r2 = 4π (3.2)2 = 4 × 3.14 × 3.2 × 3.2 = 128.6 cm2 d.) Sample Problem for Differential Volume Problem: Calculate the volume of sphere with radius 4 cm. Solution: Volume of sphere
a.) Illustration
b.) Derivation of formula The volume of cylindrical element is... 𝒅𝑽 = 𝝅𝒙𝟐 𝒅𝒚 The sum of the cylindrical elements from 0 to r is a hemisphere, twice the hemisphere will give the volume of the sphere. Thus, 𝒓 𝐕 = 𝟐𝛑 ∫𝟎 𝒙𝟐 𝒅𝒚 From the equation of the circle 𝒙𝟐 + 𝒚𝟐 = 𝒓𝟐 ; 𝒙𝟐 = 𝒓𝟐 − 𝒚𝟐 . 𝒓 𝑽 = 𝟐𝝅 ∫𝟎 (𝒓𝟐 − 𝒚𝟐 )𝒅𝒚 𝑽 = 𝟐𝝅 [𝒓𝟐 𝒚 −
𝒚𝟑
𝟑 𝒓𝟑
]
𝑽 = 𝟐𝝅 [(𝒓𝟑 − 𝟑 ) − (𝟎 − 𝑽 = 𝟐𝝅 [ 𝑽=
𝟒𝝅𝒓𝟑 𝟑
𝟐𝒓𝟑 𝟑
𝟎𝟑 𝟑
)]
]
.
c.) Sample Problem for Differential Area
Problem: Calculate the surface area of a sphere with radius 3.2 cm Solution: Surface area of sphere = 4π r2 = 4π (3.2)2 = 4 × 3.14 × 3.2 × 3.2 = 128.6 cm2 d.) Sample Problem for Differential Volume Problem: Calculate the volume of sphere with radius 4 cm. Solution: Volume of sphere
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