Question

Area of the region bounded by the curve y = |x + 1| + 1, x = –3, x = 3 and y = 0 is

विकल्प

• 8 sq units

• 16 sq units

• 32 sq units

• None of these

Answer 1

16 sq units

Explanation:

Given equation of the curves are

y = |x + 1| + 1 = ${\displaystyle \{\begin{array}{ll}(x+1)+1\text{,}& \text{If} x+1\ge 0\\ -(x+1)+1\text{,}& \text{If}x+1<0\end{array}}$

= ${\displaystyle \{\begin{array}{ll}x+2\text{,}& \text{if} x\ge -1\\ -x\text{,}& \text{if} x<-1\end{array}}$  ......(i)

x = – 3  ......(ii)

x = 3  ......(iii)

and y = 0  ......(iv)

Equation (ii) represents the line parallel to y-axis and passes through the point (–3, 0).

Equation (iii) represents the line parallel to y-axis and passes through the point (3, 0).

∴ Required area = ${\displaystyle {\int }_{-3}^{-1}y dx+{\int }_{-1}^3 y dx}$

= ${\displaystyle {\int }_{-3}^{-1}-x dx+{\int }_{-1}^3 (x+2) dx}$

= ${\displaystyle {\left[\frac{-x^2}{2}\right]}_{-3}^{-1}+{[\frac{x^2}{2}+2x]}_{-1}^3}$

= 16 sq.units