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Question

Mathematics Question on applications of integrals

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

A

16loge214316 \log _e 2-\frac{14}{3}

B

8loge21338 \log _e 2-\frac{13}{3}

C

8loge2+768 \log _e 2+\frac{7}{6}

D

16loge2+7316 \log _e 2+\frac{7}{3}

Answer

16loge214316 \log _e 2-\frac{14}{3}

Explanation

Solution

Area =1∫2​(x2−1)dx+2∫8​(x8​−1)dx
=(3x3​)12​+8(ℓnx)28​−(x)18​
=37​+8(2ℓn2)−7
=16ln2−314​

So, the correct option is (A): 16log e ​ 2−314​