The area (in sq. units) of the region bounded by the curves y = 2x and y = |x + 1|, in the first quadrant is :

Question

The area (in sq. units) of the region bounded by the curves y = 2x and y = |x + 1|, in the first quadrant is : (A) (3/2) - (1/ loge 2)  (B) (1/2) (C)  loge 2+ (3/2) (D)  (3/2)

Tardigrade Admin · Accepted Answer

Required Area ∫10 ((x+1)-2x)dx = ((x2/2)+x - (2x/In 2))10 = ((1/2) + 1 - (2/In 2)) - (0+0 - (1/In 2)) = (3/2) - (1/In 2)

Q. The area (in sq. units) of the region bounded by the curves $y = 2^{x}$ and $y = ∣ x + 1∣$ , in the first quadrant is :

$\frac{3}{2} - \frac{1}{l o g _{e} 2}$

$\frac{1}{2}$

$l o g_{e} 2 + \frac{3}{2}$

$\frac{3}{2}$

Required Area
$\int_{0}^{1} ((x + 1) - 2^{x}) d x$
$= (\frac{x ^{2}}{2} + x - \frac{2 ^{x}}{I n 2})_{0}^{1}$
$= (\frac{1}{2} + 1 - \frac{2}{I n 2}) - (0 + 0 - \frac{1}{I n 2})$
$= \frac{3}{2} - \frac{1}{I n 2}$

Q. The area (in sq. units) of the region bounded by the curves y=2x and y=∣x+1∣, in the first quadrant is :

23​−loge​21​

21​

loge​2+23​

23​