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Consider curves S_(1): sqrt(|x|)+sqrt(|y|)=sqrt(a), S_(2): x^(2)+y^(2)=a^(2) and S_(3)": "|x|+|y|=a." If "alpha" is area bounded by "S_(1) and S_(2), beta" is area bounded by "S_(1) and S_(3) and gamma is the area bounded by S_(2) and S_(3), then |
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Answer» `alpha=a^(2)(pi-(2)/(3))` Consider `x,YGT0` `therefore""sqrt(x)+sqrt(y)=sqrt(a)` `therefore""y=(sqrt(a)-sqrt(x))^(2)` `y'=2(sqrt(a)-sqrt(x))(-(1)/(sqrt(x)))=2(-(sqrt(a))/(sqrt(x)))` `therefore""y''=(sqrt(a))/(x^(3//2))gt0` So, graph is concave upward. Area bounded by `S_(1)` and coordinate AXIS in first quadrant `=overset(a)underset(0)int(sqrt(a)-sqrt(x))^(2)DX` `=overset(a)underset(0)int[a+x-2sqrt(a)sqrt(x)]dx` `=[ax+(x^(2))/(2)-4sqrt(a)(x^(3//2))/(3)]_(0)^(a)` `=a^(2)+(a^(2))/(2)-(4a^(2))/(3)` `=(a^(2))/(6)` Area bounded by `S_(2)` and coordinate axis in first quadrant `=(pia^(2))/(4)` Area bounded by `S_(3)` and coordinate axis in first quadrant `=(a^(2))/(2)` `therefore"Area bounded by "S_(1) and S_(3), alpha=4((pia^(2))/(4)-(a^(2))/(6))` `=a^(2)(pi-(2)/(3))` `"Area bounded by "S_(1) and S_(3), beta=4((a^(2))/(2)-(a^(2))/(6))=(4a^(2))/(3)` `"Area bounded by "S_(2) and S_(3),gamma=4((pia^(2))/(4)-(a^(2))/(2))=a^(2)(pi-2)` `"Also, "(beta)/(gamma)=(4)/(3sqrt((pi-2)))` |
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