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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 51-100 of 225

  1. The area of the region $\{(x,y):|x-1|\le y\le\sqrt{5-x^2}\}$ is equal to :
  2. The area of the smaller region enclosed by the curves $y^2=8x+4$ and $x^2+y^2+4\sqrt3x-4=0$ is equal to
  3. The area (in sq. units) of the region $\{(x,y):0\le y\le x^2+1,\ 0\le y\le x+1,\ \frac12\le x\le2\}$ is :
  4. The area bounded by the curve $4y^{2}=x^{2}(4-x)(x-2)$ is equal to :
  5. The area (in sq. units) of the region enclosed between the parabola $y^{2}=2x$ and the line $x+y=4$ is $\_\_\_\_\_\_$ .
  6. The area of the region bounded by the curve $y=\max\{|x|,x|x-2|\}$, the $x$-axis and the lines $x=-2$ and $x=4$ is equal to $\_\_\_\_\_\_$ .
  7. The area (in sq. units) of the part of the circle $x^{2}+y^{2}=169$ which is below the line $5x-y=13$ is …
  8. Let the area enclosed by the lines $x+y=2$, $y=0$, $x=0$ and the curve $f(x)=\min\left\{x^{2}+\frac{3}{4},\ 1+[x]\right\}$ where $[x]$ denotes the greatest integer …
  9. If the area of the region $\left\{(x,y):\left|x^{2}-2\right|\le y\le x\right\}$ is $A$, then $6A+16\sqrt{2}$ is equal to $\_\_\_\_\_\_$ .
  10. If the area of the region $S=\left\{(x,y):2y-y^{2}\le x^{2}\le2y,\ x\ge y\right\}$ is equal to $\frac{n+2}{n+1}-\frac{\pi}{n-1}$, then the natural number $n$ is equal to …
  11. The area of the region enclosed by the parabola $(y-2)^{2}=x-1$, the line $x-2y+4=0$ and the positive coordinate axes is $\_\_\_\_\_\_$ .
  12. Let the area of the region enclosed by the curve $y=\min\{\sin x,\cos x\}$ and the $x$-axis between $x=-\pi$ to $x=\pi$ be $A$. Then $A^{2}$ is equal to $\_\_\_\_\_\_$ .
  13. The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2y^{2}=kx$ and $ky^{2}=2(y-x)$ is maximum, is equal to …
  14. Let $y=p(x)$ be the parabola passing through the points $(-1,0)$, $(0,1)$ and $(1,0)$. If the area of the region …
  15. Let the area of the region $\{(x,y):x-2y+4\ge0,\ x+2y^{2}\ge0,\ x+4y^{2}\le8,\ y\ge0\}$ be $\frac{m}{n}$, where $m$ and $n$ are coprime numbers. Then $m+n$ is equal to …
  16. If for some $\alpha>0$, the area of the region $\{(x,y):|x+\alpha|\le y\le2-|x|\}$ is equal to $\frac{3}{2}$, then the area of the region …
  17. If the area of the region enclosed by the curve $y=x^{2}$ and the line $y=ax$ is $36$ sq. units, then $|a|$ is equal to $\_\_\_\_\_\_$ .
  18. Let $f:\mathbf{R}\to\mathbf{R}$ be a function such that $f(x)+3f\left(\frac{\pi}{2}-x\right)=\sin x$, $x\in\mathbf{R}$. Let the maximum value of $f$ on $\mathbf{R}$ be …
  19. Let $A$ be the region enclosed by the parabola $y^{2}=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $A$ is $\_\_\_\_\_\_$ .
  20. If the area of the region $\{(x,y):|x-5|\le y\le4\sqrt{x}\}$ is $A$, then $3A$ is equal to $\_\_\_\_\_\_$ .
  21. If the area of the region $\left\{(x,y):\left|4-x^{2}\right|\le y\le x^{2},\ y\le4,\ x\ge0\right\}$ is $\left(\frac{80\sqrt{2}}{\alpha}-\beta\right)$ …
  22. Let $f:[-3,1]\to\mathbf{R}$ be given as $f(x)=\begin{cases}\min\{(x+6),x^{2}\}, & -3\le x\le0\\ \max\{\sqrt{x},x^{2}\}, & 0\le x\le1\end{cases}$ If the area bounded by …
  23. If $A$ is the area in the first quadrant enclosed by the curve $C:2x^{2}-y+1=0$, the tangent to $C$ at the point $(1,3)$ and the line $x+y=1$, then the value of $60A$ is …
  24. If the area (in sq. units) bounded by the parabolas $5x^{2}-y=0$ and $2x^{2}-y+b=0$ ($b>0$) is $12\sqrt{3}$, then '$b$' is equal to $\_\_\_\_\_\_$ .
  25. Let $S$ be the region bounded by the curves $y=x^{3}$ and $y^{2}=x$. The curve $y=2|x|$ divides $S$ into two regions of areas $R_1$ and $R_2$. If $\max\{R_1,R_2\}=R_2$ …
  26. If the area enclosed by the curves $y=x^2$, $y^3=x$, $x=-1$ and $y=x^2$, $y^3=x$, $x=1$, above the line $x=2y$ is $\frac{n}{n+1}$, then $n$ is equal to ________.
  27. Let the area enclosed by the $x$-axis, and the tangent and normal drawn to the curve $4x^3-3xy^2+6x^2-5xy-8y^2+9x+14=0$ at the point $(-2,3)$ be $A$. Then $8A$ is equal …
  28. The area of the region $S=\{(x,y):3x^{2}\le4y\le6x+24\}$ is $\_\_\_\_\_\_$ .
  29. The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same …
  30. Let $a$ and $b$ respectively be the points of local maximum and local minimum of the function $f(x)=2x^{3}-3x^{2}-12x$. If $A$ is the total area of the region bounded by …
  31. If the line $y=mx$ bisects the area enclosed by the lines $x=0$, $y=0$, $x=\frac{3}{2}$ and the curve $y=1+4x-x^{2}$, then $12m$ is equal to $\_\_\_\_\_\_$ .
  32. Three points $O(0,0)$, $P(a,a^{2})$, $Q(-b,b^{2})$, $a>0$, $b>0$, are on the parabola $y=x^{2}$. Let $S_1$ be the area of the region bounded by the line $PQ$ and the …
  33. If the area of the region $\left\{(x,y):x^{\frac{2}{3}}+y^{\frac{2}{3}}\le1,\ x+y\ge0,\ y\ge0\right\}$ is $A$, then $\frac{256A}{\pi}$ is equal to $\_\_\_\_\_\_$ .
  34. The area bounded by the lines $y=\big||x-1|-2\big|$ is $\_\_\_\_\_\_$ .
  35. The area (in sq. units) of the region bounded by the curves $x^{2}+2y-1=0$, $y^{2}+4x-4=0$ and $y^{2}-4x-4=0$, in the upper half plane is $\_\_\_\_\_\_$ .
  36. If the area bounded by the curve $2y^{2}=3x$, lines $x+y=3$, $y=0$ and outside the circle $(x-3)^{2}+y^{2}=2$ is $A$, then $4(\pi+4A)$ is equal to $\_\_\_\_\_\_$ .
  37. The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ and the lines $x=0$, $x=\frac\pi2$, is :
  38. The area (in square units) of the region bounded by the curves $y+2x^2=0$ and $y+3x^2=1$, is equal to
  39. The area (in sq. units) of the region $\left\{(x,y)\in\mathbb R^2\mid4x^2\le y\le8x+12\right\}$ is :
  40. The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^{2}+1$, the tangent to it at the point $(2,5)$ and the coordinate axes is :
  41. The area of the region $A=\left\{(x,y):|\cos x-\sin x|\le y\le\sin x,\ 0\le x\le\frac\pi2\right\}$ is
  42. If the area enclosed between the curves $y=kx^{2}$ and $x=ky^{2},\ (k>0)$, is $1$ sq. unit. Then $k$ is
  43. The area (in sq. units) of the region $A=\left\{(x,y):x^{2}\le y\le x+2\right\}$ is
  44. Let $g(x)=\cos x^2$, $f(x)=\sqrt x$, and $\alpha,\beta$ $(\alpha<\beta)$ be the roots of the quadratic equation $18x^2-9\pi x+\pi^2=0$. Then the area (in sq. units) …
  45. The area (in sq. units) of the smaller portion enclosed between the curves $x^{2}+y^{2}=4$ and $y^{2}=3x$, is :
  46. Let $T$ and $C$ respectively, be the transverse and conjugate axes of the hyperbola $16x^2-y^2+64x+4y+44=0$. Then the area of the region above the parabola $x^2=y+4$ …
  47. The area of the region (in sq. units), enclosed by the circle $x^{2}+y^{2}=2$ which is not common to the region bounded by the parabola $y^{2}=x$ and the straight line …
  48. The area (in sq. units) of the region bounded by the curve $x^{2}=4y$ and the straight line $x=4y-2$ is :
  49. The area bounded by the curves $y=\cos x$ and $y=\sin x$ between the ordinates $x=0$ and $x=\frac{3\pi}{2}$ is
  50. The area of the region $R=\{(x,y):xy\le27,\ 1\le y\le x^{2}\}$ is equal to :

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