Question

The area made by curve f(x) = [x] + $\sqrt { \mathrm { x } - [ \mathrm { x } ] }$ and x axis when 0 £ x £ n (n Î N) is equal to {where [x] is greatest integer function}

• A. $\frac { 2 n } { 3 } + \frac { n ( n + 1 ) } { 2 }$
• B.
• C. $\frac { 2 n } { 3 } + \frac { n ( n - 1 ) } { 2 }$
• D. $\frac { \mathrm { n } } { 3 } + \frac { \mathrm { n } ( \mathrm { n } - 1 ) } { 2 }$

Answer 1

Answer: (C)

$\frac { 2 n } { 3 } + \frac { n ( n - 1 ) } { 2 }$

Answer 2

Curve

Area 0 £ x < 1 $\int _ { 0 } ^ { 1 } \sqrt { \mathrm { x } }$ dx

Area 1 £ x < 2 $\int _ { 0 } ^ { 1 } \sqrt { \mathrm { x } }$ dx + 1 × 1 (Area of A₁

A₂ A₃ A₄) Area 2 £ x < 3 $\int _ { 0 } ^ { 1 } \sqrt { \mathrm { x } }$ dx + 2 × 1

Area n – 1 £ x < n dx + (n – 1) × 1 So total

area = dx + [1 + 2 + 3 +...... ....+ (n – 1)] =