find the area of the region : {(x,y) : x2 + y2 =2ax, y2 =ax, x,y = - Maths - Application of Integrals

Question

find the area of the region : {(x,y) : x2 +
y2 <=2ax, y2 >=ax, x,y >= 0}

Ajanta Trivedi · Accepted Answer

the given regions are
x2+y2≤2ax ⇒(x-a)2+y2≤a2  
(1)which is the region inside the circle with radius a and centre as (a,0)y2≥ax which is the region outside the parabola y2=ax 
(2)x,y≥0 i e
 region lies in the 1st quadrant 
(3)
<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/content_ck_images/images/pic19_02_1%20png"
style="width: 271px; height: 220px;">
therefore the required region is
R=∫0a[2ax-x2-ax ] dx=∫0a2ax-x2 dx-a ∫0ax
dx=∫0aa2-(x-a)2 dx-a∫0ax1/2 dx=(x-a)a2-(x-a)22+a22
sin-1x-a20a-a
x3/23/20a        [since ∫a2-x2
dx=xa2-x22+a22 sin-1xaR=(a-a)a2-(a-a)22+a22
sin-1a-a2-(0-a)a2-(0-a)22-a22
sin-10-a2                           -a
23 a3/2
R=0+0-0+a22 sin-11-23a2=a22 π2-23 a2=a2
(π4-23) sq units

hope this helps you

find the area of the region : {(x,y) : x2 + y2 <=2ax, y2 >=ax, x,y >= 0}

find the area of the region : {(x,y) : x² + y²<=2ax, y² >=ax, x,y >= 0}