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Answer»  b. `y^(2)=x^(3) and |y|=2x,` both the curve are SYMMETRIC about y-axis  `4X^(2)=x^(3)or x=0, 4. ` `"Required area "=2overset(4)underset(0)int(2x-x^(3//2))dx=(32)/(5)` sq. units. c.`sqrt(x)+sqrt(|y|)=1`  The curve is symmetrical about x-axis `sqrt(|y|)=1-sqrt(x)and sqrt(x)=1-sqrt(|y|)` `rArr"for "xgt0,ygt0sqrt(y)=1-sqrt(x)` `(1)/(2sqrt(y))(DY)/(dx)=-(1)/(2sqrt(x))` `(dy)/(dx)=-sqrt((y)/(x))` `(dy)/(dx)lt0," function is decreasing "` `"Required area "=2overset(1)underset(0)int((1-x)-(1-2sqrt(x)+x))dx` `=4overset(1)underset(0)int(sqrt(x)-x)dx` `=4[(x^(3//2))/(3//2)-(x^(2))/(2)]_(0)^(1)` `=4[(2)/(3)-(1)/(2)]` `=(2)/(3)` sq. units. d. If `-8ltxlt8,` then y=2. `"If "x in (-8sqrt(2),-8]cup[8,8sqrt(2))," then "y=3,` and so on Intersection of `y=x-1 and y=2." We get "x=3 in (-8,8).` Intersection of `y=x-1 and y=3`. `"We get "x=4 notin (-8sqrt(2),-8]cup[8,8sqrt(2))`. `"Similarly, "y = x-1" will not intersect "y=[(x^(2))/(64)+2]" at any"` other integral, except in the interval `x in (-8,8).` Required area (shaded REGION ) `=2xx3-(1)/(2)xx2xx2` =4 sq. units. 
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