Integration And Area Calculation

Step 1: Rewrite the equations. $%5Ctext%7BCircle%3A%20%7D%20x%5E%7B2%7D%20%2B%20y%5E%7B2%7D%20%3D%208x%20%5C(%5CRightarrow%5C)%20(x-4)%5E%7B2%7D%20%2B%20y%5E%7B2%7D%20%3D%2016$ This is a circle with centre $(4%2C0)$ and radius $4$ . $%5Ctext%7BParabola%3A%20%7D%20y%5E%7B2%7D%20%3D%204x%20%5C(%5CRightarrow%5C)%20x%20%3D%20%5Cfrac%7By%5E%7B2%7D%7D%7B4%7D$

Step 2: Points of intersection. Substitute $x%20%3D%20%5Cfrac%7By%5E%7B2%7D%7D%7B4%7D$ in circle equation: $%5Cleft(%5Cfrac%7By%5E%7B2%7D%7D%7B4%7D%5Cright)%5E%7B2%7D%20%2B%20y%5E%7B2%7D%20%3D%208%5Cleft(%5Cfrac%7By%5E%7B2%7D%7D%7B4%7D%5Cright)$ $%5Cfrac%7By%5E%7B4%7D%7D%7B16%7D%20%2B%20y%5E%7B2%7D%20%3D%202y%5E%7B2%7D%20%5C(%5CRightarrow%5C)%20%5Cfrac%7By%5E%7B4%7D%7D%7B16%7D%20%3D%20y%5E%7B2%7D$ $y%5E%7B2%7D%5Cleft(%5Cfrac%7By%5E%7B2%7D%7D%7B16%7D%20-%201%5Cright)%20%3D%200%20%5C(%5CRightarrow%5C)%20y%20%3D%200%2C%20%5C%3B%20y%20%3D%20%5Cpm%204$ On $x$ -axis, $y%3D0$ \Rightarrow $x%3D0$ . For $y%3D4$ , $x%20%3D%20%5Cfrac%7B16%7D%7B4%7D%20%3D%204$ . So, points of intersection: $(0%2C0)$ and $(4%2C4)$ .

Step 3: Required area. We want area bounded by parabola, circle, and $x$ -axis above. In terms of $y$ : $%5Ctext%7BParabola%3A%20%7D%20x%20%3D%20%5Cfrac%7By%5E%7B2%7D%7D%7B4%7D%2C%20%5Ctext%7BCircle%3A%20%7D%20x%20%3D%204%20%2B%20%5Csqrt%7B16%20-%20y%5E%7B2%7D%7D$ Thus, area is: $A%20%3D%20%5Cint_%7B0%7D%5E%7B4%7D%20%5Cleft%5B%20%5Cbig(4%20%2B%20%5Csqrt%7B16%20-%20y%5E%7B2%7D%7D%5Cbig)%20-%20%5Cfrac%7By%5E%7B2%7D%7D%7B4%7D%20%5Cright%5D%20dy$

Step 4: Solve integration. $A%20%3D%20%5Cint_%7B0%7D%5E%7B4%7D%20%5Cleft(4%20-%20%5Cfrac%7By%5E%7B2%7D%7D%7B4%7D%5Cright)%20dy%20%2B%20%5Cint_%7B0%7D%5E%7B4%7D%20%5Csqrt%7B16%20-%20y%5E%7B2%7D%7D%20%5C%2C%20dy$ First part: $%5Cint_%7B0%7D%5E%7B4%7D%20%5Cleft(4%20-%20%5Cfrac%7By%5E%7B2%7D%7D%7B4%7D%5Cright)%20dy%20%3D%20%5Cleft%5B4y%20-%20%5Cfrac%7By%5E%7B3%7D%7D%7B12%7D%5Cright%5D_%7B0%7D%5E%7B4%7D%20%3D%2016%20-%20%5Cfrac%7B64%7D%7B12%7D%20%3D%2016%20-%20%5Cfrac%7B16%7D%7B3%7D%20%3D%20%5Cfrac%7B32%7D%7B3%7D$ Second part: $%5Cint_%7B0%7D%5E%7B4%7D%20%5Csqrt%7B16%20-%20y%5E%7B2%7D%7D%20%5C%2C%20dy$ This represents a quarter circle of radius $4$ , so area = $%5Cfrac%7B1%7D%7B4%7D%20%5Cpi%20r%5E%7B2%7D%20%3D%204%5Cpi$ . Total Area: $A%20%3D%20%5Cfrac%7B32%7D%7B3%7D%20%2B%204%5Cpi$

Final Answer: $%5Cboxed%7B%5C%3B%20%5Cdfrac%7B32%7D%7B3%7D%20%2B%204%5Cpi%20%5C%3B%7D$

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About Integration And Area Calculation - UP-BOARD-XII

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