The area (in sq. units) of the region bounded by the curve y = |x − 1| + |x + 1|, the x-axis and the lines x = −2 and x = 2 is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 10
Removing the modulus signs on each part of the interval:
\(-1 \le x \le 1:\ \ y = (1-x)+(x+1) = 2\)
\(1 \le x \le 2:\ \ y = (x-1)+(x+1) = 2x\)
Replacing x by −x leaves y unchanged, so the curve is symmetric about the y-axis, and y > 0 throughout. The area is twice the part from 0 to 2:
\( \displaystyle A = 2\left[\int_{0}^{1}2\,dx+\int_{1}^{2}2x\,dx\right] \)
\( \displaystyle = 2\left[2+\left[x^{2}\right]_{1}^{2}\right] \)
\(= 2(2+3) = 10\)