Area Under Simple Curves

Step 1: Total bounded area. The region is bounded between $x%3Dy%5E%7B2%7D$ and $x%3D4$ , symmetric about the $x$ -axis. So, $%5Ctext%7BArea%7D%20%3D%202%20%5Cint_%7B0%7D%5E%7B2%7D%20%5Cbig(4%20-%20y%5E%7B2%7D%5Cbig)%5C%2C%20dy$ $%3D%202%5Cleft%5B4y%20-%20%5Cfrac%7By%5E%7B3%7D%7D%7B3%7D%5Cright%5D_%7B0%7D%5E%7B2%7D%0A%3D%202%5Cleft(8%20-%20%5Cfrac%7B8%7D%7B3%7D%5Cright)%0A%3D%202%20%5Ccdot%20%5Cfrac%7B16%7D%7B3%7D%0A%3D%20%5Cfrac%7B32%7D%7B3%7D.$

Step 2: Half of the area. Each part should have area $%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20%5Cfrac%7B32%7D%7B3%7D%20%3D%20%5Cfrac%7B16%7D%7B3%7D.$

Step 3: Area between $x%3Dy%5E%7B2%3C%2Fstrong%3E$ and $x%3Da$ .} $%5Ctext%7BRequired%20area%7D%20%3D%202%20%5Cint_%7B0%7D%5E%7B%5Csqrt%7Ba%7D%7D%20(a%20-%20y%5E%7B2%7D)%5C%2C%20dy$ $%3D%202%5Cleft%5Bay%20-%20%5Cfrac%7By%5E%7B3%7D%7D%7B3%7D%5Cright%5D_%7B0%7D%5E%7B%5Csqrt%7Ba%7D%7D%0A%3D%202%5Cleft(a%5Csqrt%7Ba%7D%20-%20%5Cfrac%7Ba%5E%7B3%2F2%7D%7D%7B3%7D%5Cright)%0A%3D%202%5Cleft(%5Cfrac%7B2a%5E%7B3%2F2%7D%7D%7B3%7D%5Cright)%0A%3D%20%5Cfrac%7B4%7D%7B3%7Da%5E%7B3%2F2%7D.$

Step 4: Equating to half area. $%5Cfrac%7B4%7D%7B3%7Da%5E%7B3%2F2%7D%20%3D%20%5Cfrac%7B16%7D%7B3%7D$ $a%5E%7B3%2F2%7D%20%3D%204%20%5C%3B%5C%3B%20%5C(%5CRightarrow%5C)%20%5C%3B%5C%3B%20a%20%3D%204%5E%7B2%2F3%7D.$

Final Answer: $%5Cboxed%7Ba%20%3D%204%5E%7B2%2F3%7D%7D$

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

Area Under Simple Curves

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs