How do you find the volume of the region left of y = 2 x and below y=2 rotated about the y-axis?

y ≤ 2 ⇔≥ 2 x ⇔ 0 ≤ x ≤ 2So, you can use Guldino's formula for the volume of a solid of revolution: V = ( f , a lpha ) = α ∫ D f ( x ) 2 d x, where D is the domain of fSo here we have V = π ∫ 0 2 2 x d x = π x 2 | 0 2 = 4 π

Key to "How do you find the volume of the..."

High School
Calculus and Analysis
Calculus 2
Applications of integrals

telegrafyx

Answered question

2022-06-23

How do you find the volume of the region left of

y = \sqrt{2 x}

and below y=2 rotated about the y-axis?

Answer & Explanation

Hadley Cunningham

Beginner2022-06-24Added 20 answers

y \leq 2 \Leftrightarrow\geq \sqrt{2 x} \Leftrightarrow 0 \leq x \leq 2

So, you can use Guldino's formula for the volume of a solid of revolution:

V = (f, a \lpha) = α \int_{D} f (x)^{2} d x

, where D is the domain of f
So here we have

V = π \int_{0}^{2} 2 x d x = π x^{2} |_{0}^{2} = 4 π

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