Pure 2.4 - Volumes of Revolution 2

Question 1

What is the formula for a volume of revolution formed when 𝑦=f(𝑥) is rotated 2π radians about the 𝑥-axis between 𝑥=𝑎 and 𝑥=𝑏?

Answer 1

V=π∫ₐᵇ𝑦²d𝑥

Question 2

What is the formula for a volume of revolution formed when 𝑥=f(𝑦) is rotated 2π radians about the 𝑦-axis between 𝑦=𝑎 and 𝑦=𝑏?

Answer 2

V=π∫ₐᵇ𝑥²d𝑦

Question 3

What are the formulae for the volume of revolution formed by parametric curve with equations 𝑦=f(𝑡) and 𝑥=g(𝑡):
about the 𝑥-axis at 𝑥=𝑎 and 𝑥=𝑏
about the 𝑦-axis at 𝑦=𝑎 and 𝑦=𝑏

Answer 3

V=π∫ₓ₌ₐˣ⁼ᵇ𝑦²d𝑥=π∫ₜ₌ₚᵗ⁼ʳ𝑦²(d𝑥/d𝑡)d𝑡

V=π∫ᵧ₌ₐʸ⁼ᵇ𝑥²d𝑦=π∫ₜ₌ₚᵗ⁼ʳ𝑥²(d𝑦/d𝑡)d𝑡