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Answer» `{:(a,b,c,d),(q,p,s,r):}`  `"Total area "=2overset(4)underset(0)intsqrt(x)DX=(32)/(3)` `"Area of each PART "=8//3` `A_(3)=A_(4)rArroverset(4)underset(a)int(sqrt(x)-b)dx=overset(4)underset(a)int(b+sqrt(x))dx=(8)/(3)` `therefore""b=0` `therefore""overset(4)underset(a)intsqrt(x)dx=(8)/(3)` `therefore""a^(3)=16` b. `f(x)={{:(x^((1)/(log_(e)x))",",xne1),(e",",x=1):}={{:(x^((1)/(log_(e)x))",",xne1),(e",",x=1):}=e` Hence required area bound by `y=f(x) and y=|x-e|" is "e^(2)` c.  Area = Area of rectangle `OABC-overset(e)underset(1)" In "xdx` `=e-1` SQ. units. d. Solving `2 cos x =3 TAN x we GET, 2-2 sin^(2)x =3 sin x` `rArr""sin x=(1)/(2)rArrx=(pi)/(6)` `"Required area "=overset(pi//6)underset(0)int(2cos x- 3 tan x)dx` `=(2 sin x -3" In "sec x)_(0)^(pi//6)` `=1+(3)/(2)log_(e)3-3log_(e)2` |
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