1. Chapter 8 Class 12 Application of Integrals
2. Serial order wise

Transcript

Misc 11 Using the method of integration find the area bounded by the curve ๐ฅ๏ทฏ+ ๐ฆ๏ทฏ=1 [Hint: The required region is bounded by lines ๐ฅ+๐ฆ= 1, ๐ฅ โ๐ฆ=1, โ๐ฅ+๐ฆ =1 and โ๐ฅ โ๐ฆ=1 ] We know that โ๐ฅโ= ๐ฅ, ๐ฅโฅ0๏ทฎ&โ๐ฅ, ๐ฅ<0๏ทฏ๏ทฏ & โ๐ฆโ= ๐ฆ, ๐ฆโฅ0๏ทฎ&โ๐ฆ, ๐ฆ<0๏ทฏ๏ทฏ So, we can write โ๐ฅโ+โ๐ฆโ=1 as ๐ฅ+๐ฆ=1 ๐๐๐ ๐ฅ>0 , ๐ฆ>0๏ทฎโ๐ฅ+๐ฆ=1 ๐๐๐ ๐ฅ<0 ๐ฆ>0๏ทฏ๏ทฎ ๐ฅโ๐ฆ =1 ๐๐๐ ๐ฅ>0 , ๐ฆ<0๏ทฎโ๐ฅโ๐ฆ=1 ๐๐๐ ๐ฅ<0 ๐ฆ<0๏ทฏ๏ทฏ๏ทฏ๏ทฏ For ๐+๐=๐ For โ๐+๐=๐ Hence the figure is Since the Curve symmetrical about ๐ฅ & ๐ฆโ๐๐ฅ๐๐  Required Area = 4 ร Area AOB Area AOB Area ABO = 0๏ทฎ1๏ทฎ๐ฆ ๐๐ฅ๏ทฏ where ๐ฅ+๐ฆ=1 ๐ฆ=1โ๐ฅ Therefore, Area ABO = 0๏ทฎ1๏ทฎ 1โ๐ฅ๏ทฏ ๐๐ฅ๏ทฏ = ๐ฅโ ๐ฅ๏ทฎ2๏ทฏ๏ทฎ2๏ทฏ๏ทฏ๏ทฎ0๏ทฎ1๏ทฏ =1โ 1๏ทฎ2๏ทฏ๏ทฎ2๏ทฏโ 0โ 0๏ทฎ2๏ทฏ๏ทฎ2๏ทฏ๏ทฏ๏ทฎ2๏ทฏ =1โ 1๏ทฎ2๏ทฏ = 1๏ทฎ2๏ทฏ Hence, Required Area = 4 ร Area AOB = 4 ร 1๏ทฎ2๏ทฏ = 2 square units