The area of the region given by (x, y): x y ≤ 8,1 ≤ y ≤ x2 is :

Question

The area of the region given by (x, y): x y ≤ 8,1 ≤ y ≤ x2 is : (A) 16 log e 2-(14/3) (B) 8 log e 2-(13/3) (C) 8 log e 2+(7/6) (D) 16 log e 2+(7/3)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/3b37215defded402b282b44b5f517b07-.png" /> Area =∫ limits12(x2-1) d x+∫ limits28((8/x)-1) d x =((x3/3))12+8(ℓ n x)28-(x)18 =(7/3)+8(2 ℓ n 2)-7 =16 ℓ n 2-(14/3)

Q. The area of the region given by ${(x, y) : x y \leq 8, 1 \leq y \leq x^{2}}$ is :

$16 lo g_{e} 2 - \frac{14}{3}$

$8 lo g_{e} 2 - \frac{13}{3}$

$8 lo g_{e} 2 + \frac{7}{6}$

$16 lo g_{e} 2 + \frac{7}{3}$

Area $= 1 \int 2 (x^{2} - 1) d x + 2 \int 8 (\frac{8}{x} - 1) d x$
$= (\frac{x ^{3}}{3})_{1}^{2} + 8 (ℓ n x)_{2}^{8} - (x)_{1}^{8}$
$= \frac{7}{3} + 8 (2 ℓ n 2) - 7$
$= 16 ℓ n 2 - \frac{14}{3}$

Q. The area of the region given by {(x,y):xy≤8,1≤y≤x2} is :

16loge​2−314​

8loge​2−313​

8loge​2+67​

16loge​2+37​