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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 1-50 of 225

  1. Let $f:(1,\infty)\to\mathbf{R}$ be a function defined as $f(x)=\frac{x-1}{x+1}$. Let $f^{i+1}(x)=f\left(f^{i}(x)\right)$, $i=1,2,\ldots,25$, where $f^{1}(x)=f(x)$. If …
  2. The area (in sq. units) of the part of the circle $x^{2}+y^{2}=36$, which is outside the parabola $y^{2}=9x$, is :
  3. The area of the region bounded by the curves $x+3y^{2}=0$ and $x+4y^{2}=1$ is equal to:
  4. The area of the region $\{(x,y):x^{2}-8x\le y\le-x\}$ is :
  5. The area (in square units) of the region enclosed by the ellipse $x^{2}+3y^{2}=18$ in the first quadrant below the line $y=x$ is
  6. The area bounded by the curves $y=|x^2-1|$ and $y=1$ is
  7. Area of the region $\left\{(x,y):x^{2}+(y-2)^{2}\le4,\ x^{2}\ge2y\right\}$ is
  8. The area of the region enclosed by the curve $f(x)=\max\{\sin x,\cos x\}$, $-\pi\le x\le\pi$ and the $x$-axis is
  9. The area (in sq. units) of the region $\{(x,y)\in\mathbf{R}^2:x^2\le y\le3-2x\}$, is :
  10. Consider a region $R=\{(x,y)\in\mathbf{R}^2:x^2\le y\le2x\}$. If a line $y=\alpha$ divides the area of region $R$ into two equal parts, then which of the following is …
  11. The area enclosed by $y^{2}=8x$ and $y=\sqrt{2}x$ that lies outside the triangle formed by $y=\sqrt{2}x$, $x=1$, $y=2\sqrt{2}$, is equal to :
  12. The area of the region enclosed between the parabolas $y^{2}=2x-1$ and $y^{2}=4x-3$ is
  13. The area of the region enclosed by $y\le4x^2$, $x^2\le9y$ and $y\le4$, is equal to :
  14. The area of the region $\left\{(x,y):y^{2}\le4x,\ x 0,\ x\neq3\right\}$ is
  15. Given : $f(x)=\begin{cases}x, & 0\le x<\frac12\\ \frac12, & x=\frac12\\ 1-x, & \frac12<x\le1\end{cases}$ and $g(x)=\left(x-\frac12\right)^2$, $x\in\mathbf{R}$. Then the …
  16. The area of the region bounded by the parabola $(y-2)^{2}=(x-1)$, the tangent to it at the point whose ordinate is $3$ and the $x$-axis is :
  17. The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to
  18. The area (in sq. units) of the largest rectangle $ABCD$ whose vertices $A$ and $B$ lie on the $x$-axis and vertices $C$ and $D$ lie on the parabola, $y=x^2-1$ below the …
  19. The area of the region $S=\left\{(x,y):y^{2}\le8x,\ y\ge\sqrt{2}x,\ x\ge1\right\}$ is
  20. If $\int_{0}^{2}\left(\sqrt{2x}-\sqrt{2x-x^{2}}\right)dx=\int_{0}^{1}\left(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}\right)dy+\int_{1}^{2}\left(2-\frac{y^{2}}{2}\right)dy+I$ …
  21. If the area of the bounded region $R=\left\{(x,y):\max\{0,\log_ex\}\le y\le2^x,\ \frac12\le x\le2\right\}$ is $\alpha(\log_e2)^{-1}+\beta(\log_e2)+\gamma$, then the …
  22. If the area of the region $\left\{(x,y):\frac{a}{x^{2}}\le y\le\frac{1}{x},\ 1\le x\le2,\ 0<a<1\right\}$ is $(\log_e2)-\frac{1}{7}$ then the value of $7a-3$ is equal to :
  23. Area (in sq. units) of the region outside $\frac{|x|}2+\frac{|y|}3=1$ and inside the ellipse $\frac{x^2}4+\frac{y^2}9=1$ is :
  24. The area of the region bounded by $y^{2}=8x$ and $y^{2}=16(3-x)$ is equal to:
  25. The parabola $y^{2}=4x$ divides the area of the circle $x^{2}+y^{2}=5$ in two parts. The area of the smaller part is equal to :
  26. If the area of the region bounded by the curves $y=4-\frac{x^{2}}{4}$ and $y=\frac{x-4}{2}$ is equal to $\alpha$, then $6\alpha$ equals
  27. The area of the region $\left\{(x,y):x^{2}\le y\le\left|x^{2}-4\right|,\ y\ge1\right\}$ is
  28. The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to :
  29. The area of the region enclosed by the parabolas $y=4x-x^{2}$ and $3y=(x-4)^{2}$ is equal to
  30. The area (in square units) of the region bounded by the parabola $y^{2}=4(x-2)$ and the line $y=2x-8$, is :
  31. Let the locus of the centre $(\alpha,\beta)$, $\beta>0$, of the circle which touches the circle $x^2+(y-1)^2=1$ externally and also touches the $x$-axis be $L$. Then the …
  32. The area of the region : $R=\{(x,y):5x^{2}\le y\le2x^{2}+9\}$ is :
  33. The area (in sq. units) of the region, given by the set $\{(x,y)\in\mathbf{R}\times\mathbf{R}\mid x\ge0,\ 2x^{2}\le y\le4-2x\}$ is :
  34. A line passing through the point $A(-2,0)$, touches the parabola $P:y^{2}=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $AB$ …
  35. The area of the region $\{(x,y):|x-y|\le y\le4\sqrt{x}\}$ is
  36. For $a>0$, let the curves $C_1:y^2=ax$ and $C_2:x^2=ay$ intersect at origin $O$ and a point $P$. Let the line $x=b$ ($0<b<a$) intersect the chord $OP$ and the $x$-axis …
  37. The area (in sq. units) of the region enclosed by the curves $y=x^2-1$ and $y=1-x^2$ is equal to :
  38. If the area of the region $\{(x,y):1+x^{2}\le y\le\min\{x+7,11-3x\}\}$ is $A$, then $3A$ is equal to
  39. If the area of the region bounded by $16x^{2}-9y^{2}=144$ and $8x-3y=24$ is $A$, then $3(A+6\log_e(3))$ is equal to $\_\_\_\_\_\_$ .
  40. The area of the region enclosed by the curve $y=x^{3}$ and its tangent at the point $(-1,-1)$ is
  41. The area of the region bounded by $y-x=2$ and $x^2=y$ is equal to :
  42. The area enclosed by the curves $y=\log_e(x+e^2)$, $x=\log_e\left(\frac2y\right)$ and $x=\log_e2$, above the line $y=1$ is :
  43. The area of the region $\left\{(x,y):x^{2}\le y\le8-x^{2},\ y\le7\right\}$ is
  44. Let $A_1$ be the area of the region bounded by the curves $y=\sin x$, $y=\cos x$ and $y$-axis in the first quadrant. Also, let $A_2$ be the area of the region bounded by …
  45. The odd natural number $a$, such that the area of the region bounded by $y=1$, $y=3$, $x=0$, $x=y^a$ is $\frac{364}{3}$, is equal to :
  46. The area of the region in the first quadrant inside the circle $x^{2}+y^{2}=8$ and outside the parabola $y^{2}=2x$ is equal to :
  47. The area bounded by the curve $y=|x^{2}-9|$ and the line $y=3$ is :
  48. The area of the bounded region enclosed by the curve $y=3-\left|x-\frac{1}{2}\right|-|x+1|$ and the $x$-axis is :
  49. The area of the region given by $A=\{(x,y):x^2\le y\le\min\{x+2,4-3x\}\}$ is :
  50. Let the area of the region enclosed by the curves $y=3x$, $2y=27-3x$ and $y=3x-x\sqrt{x}$ be $A$. Then $10A$ is equal to

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