volume-of-solids-generated-by-revolving-the-region-bounded-by-the-curve-and-line-the-y-axis-y-x-y-x-2-x-2-

Question Number 215280 by universe last updated on 02/Jan/25

v o l u m e o f s o l i d s g e n e r a t e d b y r e v o l v i n g

t h e r e g i o n b o u n d e d b y t h e c u r v e a n d l i n e

t h e y - a x i s

y = x, y = x / 2, x = 2

Commented by mr W last updated on 02/Jan/25

r o t a t e d a b o u t w h i c h a x i s ?

Commented by universe last updated on 02/Jan/25

y - a x i s

Answered by MrGaster last updated on 02/Jan/25

V = π \int_{a}^{b} {[f (x)]}^{2} - {[g (x)]}^{2} d x

V = π \int_{0}^{2} {[x]}^{2} - {[\frac{π}{2}]}^{2} d x

V = π \int_{0}^{2} x^{2} - \frac{x^{2}}{4} d x

V = π \int_{0}^{2} \frac{4 x^{2}}{4} - \frac{x^{2}}{4} d x

V = π \int_{0}^{2} \frac{3 x^{2}}{4} d x

V = π {[\frac{3 x^{3}}{12}]}_{0}^{2}

V = π (\frac{3 \cdot 2^{3}}{12} - \frac{3 \cdot 0^{3}}{12})

V = π (\frac{3 \cdot 8}{12})

V = π (\frac{24}{12})

V = 2 π

Answered by mr W last updated on 02/Jan/25

i f r o t a t e d a b o u t y - a x i s :

V = 2 π \times \frac{2 \times 2}{3} \times \frac{2 \times 1}{2} = \frac{8 π}{3}

i f r o t a t e d a b o u t x - a x i s :

V = 2 π (\frac{2}{3} \times \frac{2 \times 2}{2} - \frac{1}{3} \times \frac{2 \times 1}{2}) = 2 π

Commented by universe last updated on 02/Jan/25

w i t h o u t i n t e g r a l s i r

c a n u e x p l a i n l i t t l e b i t

Commented by mr W last updated on 02/Jan/25

V_{a b o u t y - a x i s} = \int_{A} 2 π x d A = 2 π x_{S} A

V_{a b o u t x - a x i s} = \int_{A} 2 π y d A = 2 π y_{S} A

(x_{S}, y_{S}) = c e n t e r o f a r e a

s e e a l s o Q 158237

Commented by universe last updated on 02/Jan/25

t h a n k s