1.

Given function `f(x) ={:{(x^(2),"for " 0 lexlt1),(sqrt(x),"for "1 le x le2):}, "then" int_(0)^(2)f(x)dx` is equal toA. ` (4sqrt(2)-1)`B. `1/3(4sqrt(2)-1)`C. ` 1/3 (sqrt(2)-1)`D. None of these

Answer» Correct Answer - B
Here, ` int _(0)^(pi)f(x) dx = int _(0)^(1) f(x) dx + int _(1)^(2) f(x) dx `
` :. int _(0)^(2) f(x) dx = int_(0)^(1) x^(2) sqrt(x) dx`
` rArr int _(0)^(2) f(x) dx = [ (x^(3))/3]_(0)^(1)+ [ 2/3 x sqrt(x)]_(1)^(2)`
` = [ 1/3 - 0 ] +2/3 [ 2 sqrt(2)-1]`
` = 1/3 + (4sqrt(2))/2 - 2/3 = (4sqrt(2))/3 -1/3 `


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