1 Find the area of the region {(x,y):4x2+4y2 =9 and y2 =4x} 2 Find the area lying above - Maths - Application of Integrals

Question

1 Find the area of the region {(x,y):4x2
​+4y2 ​<=9 and
y2<=4x}
​2 Find the area lying above x-axis included between the
circle x2+y2=8x and y2=4x
3 Using integration, find the area of the region bounded by the
line x - y+2=0 and curve x=root of y and y - axis

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

2 We havex2+y2=8x⇒y2=8x-x2
iy1=8x2-x2 Take + sign because we need area above x-axisy2=4xy2=2x
iiNow let us find find point of intersection8x-x2=4x⇒x2-4x=0⇒x=0 and x=4Hence desired area is given by:A=∫04y1-y2
dx=∫048x-x2-2x dx=∫048x-x2-2∫04x
dx=∫04-x2-8x-2∫04x dx=∫04-x-42-16-2∫04x
dx=∫0416-x-42-2∫04x dx=x-42
16-x-42+162sin-1x-44-2  x323204=x-42
16-x-42+162sin-1x-44-43  x3204=x-42
16-x-42+8sin-1x-44-43  x3204=0+0-43×8-0-42
16-0-42+8sin-10-44-0=-32-8×-π2=-323+4π=4π-323 AnswerPlease ask one question in one thread

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1. Find the area of the region {(x,y):4x2 ​+4y2 ​<=9 and y2<=4x} ​2. Find the area lying above x-axis included between the circle x2+y2=8x and y2=4x 3.Using integration, find the area of the region bounded by the line x - y+2=0 and curve x=root of y and y - axis.

1. Find the area of the region {(x,y):4x² +4y² <=9 and y²<=4x}
2. Find the area lying above x-axis included between the circle x²+y²=8x and y²=4x
3.Using integration, find the area of the region bounded by the line x - y+2=0 and curve x=root of y and y - axis.