Question

(1 point) Find the volume of the solid obtained by rotating the region in the first quadrant bounded by \(y=x^{7}, y=1\), and the \(y\) -axis around the \(y\) -axis.

Volume =

Answer 1

The statement of disk method/ washer method when region is revolved about Y axis is:

Disk method / washer was her method / Revolution about Y anie ) X are integrable Let flufa : [a,b] > R functi off Let region ion such that 14 fa X= is between curven Linea bounded fily), 2e = fa(y) and between y = 0, y=b. volume of of solid solid generated when region i revolued about Y-axin in Then d 2 2 V IT (facy))–(fory) di y=C

Now let's begin to solve given question as:

in first (0) Find the volume of solid olid when region quadrant in bounded by y=x y=x7, y=1 and Yaxin about Η ακτλ. Soln! let R be region bounded by given curved firat will eketch R. Y Axix we 7 → Y=P Region R- Gy=1 x Axin we ou can exprean R R = S(x,y) : R={(x,y) ac ocx< OLY ZI For understanding purpose if me we draw a Horizontal MR. linea then it starto at Y axin (ie x=o) and enda y = xy lie rezy 7 oL x 2 y on

and Region in bounded between y=0 f g=1 OLYEI } want to we DiAK method, we select on yecoil since we व faly)= y fily) = 0 2 -10 on yf Coil since У 기 faly) → on y elo,l] y fiey) when R PA volume Now Revolution of Revolved about Y axia method ) V= T ((uring washer / Dink) is I'll ý )? -0% I dy = (y* -o) y=0 dy y=0 7 17 dy y y=0 Formula i using 6 a "I 11 xh da nal x nal a a nu-1

2+7 C → 7 "I cs I. " 247 7 ت له whace FIT y [*]. P.14 V = Volume 7T 11-0] - l volume FIT 9 Anwen or volume 2.44 35

I am attaching the graphs drawn in desmos for better understanading

III (0.1) (1.1) -10 -5 o 5 10 (0,0) -5 + Х y=x? Х X=0 -10 10 3 y=1 -10 10 4

Note: for any query or explanataion in any step, kindly mention in comment section. I will assist you ASAP.