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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 101-150 of 225

  1. Let $e$ be the base of natural logarithm and let $f:\{1,2,3,4\}\to\{1,e,e^{2},e^{3}\}$ and $g:\{1,e,e^{2},e^{3}\}\to\left\{1,\frac12,\frac13,\frac14\right\}$ be two …
  2. The area of the region $\{(x,y):0\le y\le6-x,\ y^{2}\ge4x-3,\ x\ge0\}$ is :
  3. Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, $x$-axis and the ordinates $x=\frac\pi4$ and $x=\beta>\frac\pi4$ is …
  4. The area bounded by the curve $y=\ln(x)$ and the lines $y=0$, $y=\ln(3)$ and $x=0$ is equal to :
  5. The area (in square units) bounded by the curves $y=\sqrt x$, $2y-x+3=0$, $x$-axis, and lying in the first quadrant is :
  6. The area of the region bounded by the curves $y=|x-2|$, $x=1$, $x=3$ and the $x$-axis is :
  7. The area of the region described by $A=\{(x,y):x^{2}+y^{2}\le1\text{ and }y^{2}\le1-x\}$ is :
  8. The parabolas $y^{2}=4x$ and $x^{2}=4y$ divide the square region bounded by the lines $x=4$, $y=4$ and the coordinate axes. If $S_1,S_2,S_3$ are respectively the areas …
  9. The area of the region (in sq. units), in the first quadrant bounded by the parabola $y=9x^{2}$ and the lines $x=0$, $y=1$ and $y=4$, is :
  10. The area under the curve $y=|\cos x-\sin x|$, $0\le x\le\frac\pi2$, and above $x$-axis is :
  11. The area of the region bounded by the parabola $(y-2)^{2}=x-1$, the tangent of the parabola at the point $(2,3)$ and the $x$-axis is :
  12. The area of the plane region bounded by the curves $x+2y^{2}=0$ and $x+3y^{2}=1$ is equal to :
  13. The area of the region bounded by the curves $y=|x-1|$ and $y=3-|x|$ is :
  14. The area of the region $\{(x,y):y\le\pi-|x|,\ y\le|x\sin x|,\ y\ge0\}$ is :
  15. The area of the region enclosed by the curves $y=x$, $x=e$, $y=\frac1x$ and the positive $x$-axis is :
  16. The area enclosed between the curves $y^{2}=x$ and $y=|x|$ is :
  17. The area (in sq. units) of the region described by $\{(x,y):y^{2}\le2x\text{ and }y\ge4x-1\}$ is :
  18. The area enclosed between the curve $y=\log_e(x+e)$ and the coordinate axes is :
  19. The shortest distance between the line $y=x$ and the curve $y^{2}=x-2$ is :
  20. The area (in sq. units) of the region $A=\left\{(x,y):\frac{y^{2}}{2}\le x\le y+4\right\}$ is :
  21. The area (in sq. units) of the region $\{(x,y):x\ge0,\ x+y\le3,\ x^2\le4y\text{ and }y\le1+\sqrt x\}$ is :
  22. Let $A=\left\{(x,y):y^{2}\le4x,\ y-2x\ge-4\right\}$. The area of the region $A$ in square units is
  23. The area of the region (in square units) above the $x$-axis bounded by the curve $y=\tan x$, $0\le x\le\frac\pi2$ and the tangent to the curve at $x=\frac\pi4$ is
  24. The area bounded by the curves $y=\ln x$, $y=\ln|x|$, $y=|\ln x|$ and $y=|\ln|x||$ is :
  25. The area (in sq. units) of the region $A=\{(x,y)\in R\times R\mid0\le x\le3,\ 0\le y\le4,\ y\le x^{2}+3x\}$ is
  26. Let $S(\alpha)=\{(x,y):y^2\le x,\ 0\le x\le\alpha\}$ and $A(\alpha)$ is the area of the region $S(\alpha)$. If for a $\lambda$, $0<\lambda<4$, $A(\lambda):A(4)=2:5$ …
  27. If the area (in sq. units) bounded by the parabola $y^{2}=4\lambda x$ and the line $y=\lambda x$, $\lambda>0$, is $\frac19$, then $\lambda$ is equal to
  28. The area (in sq. units) of the region bounded by the curves $y=2^{x}$ and $y=x+1$, in the first quadrant is
  29. Let $n\geq2$ be a natural number and $f:[0,1]\to\mathbb{R}$ be the function defined by $$f(x)=\begin{cases} n(1-2nx) & \text{if } 0\le x\le\frac{1}{2n}\\ 2n(2nx-1) & …
  30. Let $S=\left\{(x,y)\in\mathbb{R}\times\mathbb{R}: x\ge 0,\ y\ge 0,\ y^2\le 4x,\ y^2\le 12-2x \text{ and } 3y+\sqrt8\,x\le 5\sqrt8\right\}$. If the area of the region $S$ …
  31. The area (in sq. units) of the region $\{(x,y): y^{2}\ge2x \text{ and } x^{2}+y^{2}\le4x,\ x\ge0,\ y\ge0\}$ is
  32. The area (in sq. units) of the region described by $A=\{(x,y)\mid y\ge x^{2}-5x+4,\ x+y\ge1,\ y\le0\}$ is
  33. Let $\Delta$ be the area of the region $\left\{(x,y)\in R^{2}:x^{2}+y^{2}\le21,\ y^{2}\le4x,\ x\ge1\right\}$. Then …
  34. Consider the functions $f,\ g:\mathbb{R}\to\mathbb{R}$ defined by $$f(x)=x^{2}+\frac{5}{12}\quad\text{and}\quad g(x)=\begin{cases}2\left(1-\frac{4|x|}{3}\right), & …
  35. Let $A=\left\{(x,y)\in R^{2}:y\ge0,\ 2x\le y\le\sqrt{4-(x-1)^{2}}\right\}$ and $B=\left\{(x,y)\in R\times R:0\le y\le\min\left\{2x,\sqrt{4-(x-1)^{2}}\right\}\right\}$. …
  36. Let $f:[0,1]\to[0,1]$ be the function defined by $f(x)=\frac{x^3}{3}-x^2+\frac59x+\frac{17}{36}$. Consider the square region $S=[0,1]\times[0,1]$. Let …
  37. The area (in sq. units) of the region bounded by the parabola $y=x^{2}+2$ and the lines $y=x+1$, $x=0$ and $x=3$, is
  38. If the area of the region bounded by the curves $y=x^{2}$, $y=\frac1x$ and the lines $y=0$ and $x=t$ ($t>1$) is $1$ sq. unit, then $t$ is equal to
  39. The area (in sq. units) bounded by the parabola $y=x^2-1$, the tangent at the point $(2,3)$ to it and the $y$-axis is
  40. The area of the region $A=\{(x,y):0\le y\le x|x|+1\text{ and }-1\le x\le1\}$ in sq. units, is
  41. The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabola $y=12-x^{2}$ such that the rectangle lies …
  42. If the area (in sq. units) of the region $\{(x,y):y^2\le4x,\ x+y\le1,\ x\ge0,\ y\ge0\}$ is $a\sqrt2+b$, then $a-b$ is equal to
  43. The area of the region $\{(x,y): xy\leq 8,\ 1\leq y\leq x^2\}$ is
  44. The area enclosed between the curves $y^{2}+4x=4$ and $y-2x=2$ is
  45. The area (in sq. units) of the region $\{x\in\mathbb R: x\ge0,\ y\ge0,\ y\ge x-2\text{ and }y\le\sqrt x\}$, is
  46. Let the area enclosed between the curves $|y|=1-x^{2}$ and $x^{2}+y^{2}=1$ be $\alpha$. If $9\alpha=\beta\pi+\gamma$; $\beta,\gamma$ are integers, then the value of …
  47. Let the area of the region $\{(x,y):2y\le x^2+3,\ y+|x|\le3,\ y\ge|x-1|\}$ be $A$. Then $6A$ is equal to :
  48. Consider the region $R=\left\{(x,y): x\le y\le9-\frac{11}{3}x^{2},\ x\ge0\right\}$. The area, of the largest rectangle of sides parallel to the coordinate axes and …
  49. The area of the region $\left\{(x,y): x^{2}+4x+2\le y\le|x+2|\right\}$ is equal to
  50. Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^{2}-px+\frac54p=0$ are rational. Then the area of the region …

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