The area of the region given by { ( x , y ) : x y 8 , 1 y x 2 } is [2023]

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Q.
The area of the region given by ${(x, y) : x y \leq 8, 1 \leq y \leq x^{2}}$ is [2023]

1 $16 \log_{e} 2 - \frac{14}{3}$

2 $8 \log_{e} 2 - \frac{13}{3}$

3 $8 \log_{e} 2 + \frac{7}{6}$

4 $16 \log_{e} 2 + \frac{7}{3}$

Ans.
(1)

$Area = \int_{1}^{2} (x^{2} - 1) d x + \int_{2}^{8} (\frac{8}{x} - 1) d x$

$= {(\frac{x^{3}}{3} - x)}_{1}^{2} + 8 {(\log_{e} x)}_{2}^{8} - {(x)}_{2}^{8}$

$= \frac{4}{3} + 8 (2 \log_{e} 2) - 6 = 16 \log_{e} 2 - \frac{14}{3}$

Solution

Q.
The area of the region given by ${(x, y) : x y \leq 8, 1 \leq y \leq x^{2}}$ is [2023]

1 $16 \log_{e} 2 - \frac{14}{3}$

2 $8 \log_{e} 2 - \frac{13}{3}$

3 $8 \log_{e} 2 + \frac{7}{6}$

4 $16 \log_{e} 2 + \frac{7}{3}$

Ans.
(1)

$Area = \int_{1}^{2} (x^{2} - 1) d x + \int_{2}^{8} (\frac{8}{x} - 1) d x$

$= {(\frac{x^{3}}{3} - x)}_{1}^{2} + 8 {(\log_{e} x)}_{2}^{8} - {(x)}_{2}^{8}$

$= \frac{4}{3} + 8 (2 \log_{e} 2) - 6 = 16 \log_{e} 2 - \frac{14}{3}$

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Solution

Q. The area of the region given by {(x,y):xy≤8, 1≤y≤x2} is [2023]

1 16loge2-143

2 8loge2-133

3 8loge2+76

4 16loge2+73

Ans. (1) Area=∫12(x2-1)dx+∫28(8x-1)dx =(x33-x)12+8(logex)28-(x)28 =43+8(2loge2)-6=16loge2-143

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Q.
The area of the region given by ${(x, y) : x y \leq 8, 1 \leq y \leq x^{2}}$ is [2023]

1 $16 \log_{e} 2 - \frac{14}{3}$

2 $8 \log_{e} 2 - \frac{13}{3}$

3 $8 \log_{e} 2 + \frac{7}{6}$

4 $16 \log_{e} 2 + \frac{7}{3}$

Ans.
(1)

$Area = \int_{1}^{2} (x^{2} - 1) d x + \int_{2}^{8} (\frac{8}{x} - 1) d x$

$= {(\frac{x^{3}}{3} - x)}_{1}^{2} + 8 {(\log_{e} x)}_{2}^{8} - {(x)}_{2}^{8}$

$= \frac{4}{3} + 8 (2 \log_{e} 2) - 6 = 16 \log_{e} 2 - \frac{14}{3}$