JEE Mains · Maths · STD 12 - 8. Application and integration
The area of the region given by \(\left\{(x, y): x y \leq 8,1, \leq y \leq x^2\right\}\) is :
- A \(8 \log _e 2-\frac{13}{3}\)
- B \(16 \log _{ e } 2-\frac{14}{3}\)
- C \(8 \log _e 2+\frac{7}{6}\)
- D \(16 \log _{ e } 2+\frac{7}{3}\)
Answer & Solution
Correct Answer
(B) \(16 \log _{ e } 2-\frac{14}{3}\)
Step-by-step Solution
Detailed explanation
Area \(=\int \limits_1^2\left(x^2-1\right) d x+\int \limits_2^8\left(\frac{8}{x}-1\right) d x\) \(=\left(\frac{x^3}{3}\right)_1^2+8(\ell \operatorname{nx})_2^8-(x)_1^8\) \(=\frac{7}{3}+8(2 \ell n 2)-7\) \(=16 \ell n 2-\frac{14}{3}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let three real numbers \(a, b, c\) be in arithmetic progression and \(\mathrm{a}+1, \mathrm{~b}, \mathrm{c}+3\) be in geometric progression. If \(\mathrm{a}>10\) and the arithmetic mean of \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) is \(8\) , then the cube of the geometric mean of \(a, b\) and \(c\) isJEE Mains 2024 Medium
- Consider three observations \(a, b\) and \(c\) such that \(b = a + c .\) If the standard deviation of \(a +2\) \(b +2, c +2\) is \(d ,\) then which of the following is true ?JEE Mains 2021 Medium
- If \(\frac{1^3+2^3+3^3+\ldots \ldots \text {.upto } n \text { terms }}{1 \cdot 3+2 \cdot 5+3 \cdot 7+\ldots \ldots \text { upto } n \text { terms }}=\frac{9}{5}\), then the value of \(n\) isJEE Mains 2023 Hard
- The set of all values of \(t \in R\), for which the matrix \(\left[\begin{array}{ccc}e^t & e^{-t}(\sin t-2 \cos t) & e^{-t}(-2 \sin t-\cos t) \\e^t & e^{-t}(2 \sin t+\cos t) & e^{-t}(\sin t-2 \cos t) \\e^t & e^{-t} \cos t & e^{-t} \sin t \end{array}\right]\) is invertible.JEE Mains 2023 Hard
- Let \(10\) vertical poles standing at equal distances on a straight line, subtend the same angle of elevation at a point \(O\) on this line and all the poles are on the same side of \(O\). If the height of the longest pole is \('h'\) and the distance of the foot of the smallest pole from \(O\) is \('a'\); then the distance between two consecutive poles, isJEE Mains 2015 Hard
- \(\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2} \int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos \left(\frac{1}{t^3}\right) d t\right)\) is equal toJEE Mains 2024 Hard
More PYQs from JEE Mains
- The value of the integral \(\int\limits_0^1 {x\,{{\cot }^{ - 1}}\,\left( {1 - {x^2} + {x^4}} \right)} dx\) isJEE Mains 2019 Hard
- If the value of real number \(a > 0\) for which \(x^2-5 a x\) \(+1=0\) and \(x^2-a x-5=0\) have a common real roots is \(\frac{3}{\sqrt{2 \beta}}\) then \(\beta\) is equal toJEE Mains 2023 Hard
- If the system of linear equations \(2 x+y-z=3\) \(x-y-z=\alpha\) \(3 x+3 y+\beta z=3\) has infinitely many solution, then \(\alpha+\beta-\alpha \beta\) is equal to .... .JEE Mains 2021 Medium
- The sum of the real roots of the equation \(\left| {\begin{array}{*{20}{c}}
x&{ - 6}&{ - 1}\\
2&{ - 3x}&{x - 3}\\
{ - 3}&{2x}&{x = 2}
\end{array}} \right| = 0\) is equal toJEE Mains 2019 Hard - Let \(A\) be a matrix of order \(3 \times 3\) and \(|A|=5\). If \(|2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in \mathrm{N}\) then \(\alpha+\beta+\gamma\) is equal toJEE Mains 2025 Medium
- Let \(A\) be a point on the line \(\vec r = \left( {1 - 3\mu } \right)\hat i + \left( {\mu - 1} \right)\hat j + \left( {2 + 5\mu } \right)\hat k\) and \(B(3, 2, 6)\) be a point in the space. Then the value of \(\mu \) for which the vector \(\overrightarrow {AB} \) is parallel to the plane \(x -4y +3z = 1\) isJEE Mains 2019 Hard