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Yukti Vault

Calculus

Application of Integrals: JEE Main & Advanced PYQs

225 previous year questions on Application of Integrals from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 201-225 of 225

  1. The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0,\frac{\pi}{2}\right]$ is
  2. A farmer $F_1$ has a land in the shape of a triangle with vertices at $P(0,0)$, $Q(1,1)$ and $R(2,0)$. From this land, a neighbouring farmer $F_2$ takes away the region …
  3. Let the functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)=e^{x-1}-e^{-|x-1|}\ \text{ and }\ …
  4. If the area of the region $\{(x,y):-1\le x\le 1,\ 0\le y\le a+e^{|x|}-e^{-x},\ a>0\}$ is $\frac{e^2+8e+1}{e}$, then the value of $a$ is :
  5. The area (in sq. units) of the region $\{(x,y): 0\le y\le 2|x|+1,\ 0\le y\le x^2+1,\ |x|\le 3\}$ is
  6. If $\alpha$ is the area of the common region that lies inside both the given ellipses, then the value of $\cot\alpha$ is ____________.
  7. Let $\beta$ be the area of the region enclosed between the curves $C_1$, $C_2$, and the lines $x=\alpha_1$ and $x=\alpha_4$. Then the value of …
  8. The area of the region bounded by the curve $y=f(x)$, the $x$-axis, and the lines $x=a$ and $x=b$, where $-\infty<a<b<-2$, is
  9. The area bounded by the curve $y=f(x)$ and the lines $x=0$, $y=0$ and $x=t$, lies in the interval
  10. The area bounded between the parabolas $x^2=\frac y4$ and $x^2=9y$, and the straight line $y=2$ is :
  11. For any real $t$, $x=\dfrac{e^{t}+e^{-t}}{2}$, $y=\dfrac{e^{t}-e^{-t}}{2}$ is a point on the hyperbola $x^{2}-y^{2}=1$. Show that the area bounded by this hyperbola and …
  12. Let $A_n$ be the area bounded by the curve $y=(\tan x)^{n}$ and the lines $x=0$, $y=0$, and $x=\dfrac{\pi}{4}$. Prove that for $n>2$, $A_n+A_{n-2}=\dfrac{1}{n-1}$ and …
  13. One of the points of intersection of the curves $y=1+3x-2x^{2}$ and $y=\dfrac{1}{x}$ is $\left(\dfrac{1}{2},2\right)$. Let the area of the region enclosed by these …
  14. The area (in sq. units) of the region described by $\left\{(x,y):y^{2}\le2x,\ \text{and}\ y\ge4x-1\right\}$ is
  15. The area of the region enclosed by the parabolas $y=x^{2}-5x$ and $y=7x-x^{2}$ is $\_\_\_\_\_\_\_\_$ .
  16. The area enclosed between the curves $y=x|x|$ and $y=x-|x|$ is :
  17. Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}+\dfrac{2x}{\left(1+x^{2}\right)^{2}}y=xe^{\frac{1}{1+x^{2}}}$; $y(0)=0$. Then the area enclosed …
  18. The area (in sq. units) of the region $A=\left\{(x,y):(x-1)[x]\le y\le2\sqrt{x},\ 0\le x\le2\right\}$, where $[t]$ denotes the greatest integer function, is :
  19. Let $f:\mathbf{R}\to\mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbf{R}$. If $f'(0)=4a$ and $f$ satisfies …
  20. The area of the region, inside the circle $\left(x-2\sqrt{3}\right)^{2}+y^{2}=12$ and outside the parabola $y^{2}=2\sqrt{3}x$ is :
  21. Let the function, $f(x)=\begin{cases}-3ax^{2}-2, & x 1$, $b\in\mathbf{R}$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta\sqrt{3}$ …
  22. The area of the region enclosed by the curves $y=x^{2}-4x+4$ and $y^{2}=16-8x$ is :
  23. If the area of the larger portion bounded between the curves $x^{2}+y^{2}=25$ and $y=|x-1|$ is $\dfrac{1}{4}(b\pi+c)$, $b,c\in\mathbf{N}$, then $b+c$ is equal to …
  24. The area of the region enclosed by the curves $y=e^{x}$, $y=\left|e^{x}-1\right|$ and $y$-axis is :
  25. The area of the region bounded by the curves $x\left(1+y^{2}\right)=1$ and $y^{2}=2x$ is:

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