Calculus – Mathematics Optional PYQs (2008–2026)

Here are all 107 Calculus questions asked in UPSC Mathematics Optional Paper I from 2008 to 2026, arranged under the 5 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

2.1] Limits, continuity and differentiability

20 questions

1. Using Maclaurin’s theorem, obtain the first four terms of the expansion of $e^{xcosx}$ in ascending powers of $x$. Hence or otherwise find the limit of $\frac{e^x-e^{xcosx}}{x-sinx}$ as $x$ tends to zero. [2026/10m/1d]

2. Given that $f(x+y)=f\left(x\right)f\left(y\right)$ for all real $x$, $y$, $f\left(x\right)\ne 0$ for any real $x$ and $f\prime \left(0\right)=2$. Show that for all real $x$, $f\prime \left(x\right)=2f\left(x\right)$. Hence find $f\left(x\right)$. [2025/10m/1d]

3. Using Mean Value Theorem, prove that $\frac{\pi }{6}+\frac{\sqrt{3}}{15}<{sin}^{-1}\left(\frac{3}{5}\right)<\frac{\pi }{6}+\frac{1}{8}.$ [2025/15m/2b]

4. Discuss the continuity of the function $f\left(x\right)=\{\begin{array}{cc}\hfill \frac{1}{1-e^{-1/x}},\hfill & \hfill x\ne 0\hfill \\ \hfill 0,\hfill & \hfill x=0\hfill \end{array}$ for all values of $x$. [2024/10m/1c]

5. Expand $ln\left(x\right)$ in powers of $(x-1)$ by Taylor’s theorem and hence find the value of $ln\left(1.1\right)$ correct up to four decimal places. [2024/10m/1d]

6. Find the values of $p$ and $q$ for which ${lim}_{x\to 0}\frac{x(1+pcosx)-qsinx}{x^3}$ exists and equals 1. [2023/10m/1c]

7. Evaluate ${lim}_{x\to \infty }{(e^x+x)}^{\frac{1}{x}}$. [2022/10m/1c]

8. Given : $\Delta \left(x\right)=\mid \begin{array}{ccc}\hfill f(x+\alpha )\hfill & \hfill f(x+2\alpha )\hfill & \hfill f(x+3\alpha )\hfill \\ \hfill f\left(\alpha \right)\hfill & \hfill f\left(2\alpha \right)\hfill & \hfill f\left(3\alpha \right)\hfill \\ \hfill f\prime \left(\alpha \right)\hfill & \hfill f\prime \left(2\alpha \right)\hfill & \hfill f\prime \left(3\alpha \right)\hfill \end{array}\mid$ where $f$ is a real valued differentiable function and $\alpha$ is a constant. Find ${lim}_{x\to 0}\frac{\Delta \left(x\right)}{x}$. [2021/10m/1c]

9. Show that between any two roots of $e^{x}cosx=1$, there exists at least one root of $e^{x}sinx-1=0$. [2021/10m/1d]

10. Evaluate ${lim}_{x\to \frac{\pi }{4}}{(tanx)}^{tan2x}$. [2020/10m/1c]

11. Let $f:[0,\frac{\pi }{2}]\to ℝ$ be a continuous function such that $f\left(x\right)=\frac{{cos}^{2}x}{4x^2-{\pi }^2},0\le x<\frac{\pi }{2}$ Find the value of $f\left(\frac{\pi }{2}\right)$. [2019/10m/1a]

12. Is $f\left(x\right)=|cosx|+|sinx|$ differentiable at $x=\frac{\pi }{2}$? If yes, then find its derivative at $x=\frac{\pi }{2}$. If no, then give a proof of it. [2019/15m/2a]

13. Determine if ${lim}_{z\to 1}(1-z)tan\frac{\pi z}{2}$ exists or not. If the limit exists, then find its value. [2018/10m/1c]

14. Evaluate the following limit : ${lim}_{x\to a}{(2-\frac{x}{a})}^{tan\left(\frac{\pi x}{2a}\right)}.$ [2015/10m/1c]

15. Prove that between two real roots of $e^{x}cosx+1=0$, a real root of $e^{x}sinx+1=0$ lies. [2014/10m/1c]

16. Let $f$ be a function defined on $ℝ$ such that $f\left(0\right)=-3$ and $f\prime \left(x\right)\le 5$ for all values of $x$ in $ℝ$. How large can $f\left(2\right)$ possibly be ? [2011/10m/1d]

17. Evaluate : ${lim}_{x\to 2}f\left(x\right)$, where $f\left(x\right)=\{\begin{array}{cc}\hfill \frac{x^2-4}{x-2},\hfill & \hfill x\ne 2\hfill \\ \hfill \pi ,\hfill & \hfill x=2\hfill \end{array}$ [2011/8m/3a(i)]

18. A twice-differentiable function $f\left(x\right)$ is such that $f\left(a\right)=0=f\left(b\right)$ and $f\left(c\right)>0$ for $a<c<b$. Prove that there is at least one point $\xi$, $a<\xi <b$, for which $f″\left(\xi \right)<0$. [2010/12m/1c]

19. Suppose that $f″$ is continuous on $[1,2]$ and that $f$ has three zeroes in the interval $(1,2)$. Show that $f″$ has at least one zero in the interval $(1,2)$. [2009/12m/1c]

20. Find the value of ${lim}_{x\to 1}(1-x)cot\frac{\pi x}{2}$. [2008/12m/1c]

2.2] Maxima, minima, asymptotes and curve tracing

14 questions

1. If the sum of the lengths of the hypotenuse and another side of a right-angled triangle be given, then find the angle between these sides so that the area of the triangle is maximum. [2026/15m/2b]

2. Trace the curve $y^2(a+x)=x^2(3a-x)$. [2026/15m/4b]

3. A rectangular sheet of metal of length 6 meters and width 2 meters is given. Four equal squares are removed from the four corners. The sides of this sheet are now folded up to form an open rectangular box. Find approximately the height of the box, such that the volume of the box is maximum. [2025/10m/1c]

4. Find the volume of the greatest cylinder which can be inscribed in a cone of height $h$ and semi-vertical angle $\alpha$. [2024/20m/3b]

5. Trace the curve $y^2(x^2-1)=2x-1$. [2023/20m/4b]

6. Trace the curve $y^2{x}^2=x^2-a^2$, where $a$ is a real constant. [2022/20m/4b]

7. Find all the asymptotes of the curve $(2x+3)y={(x-1)}^2$. [2020/10m/1d]

8. Consider the function $f\left(x\right)={\int }_0^x(t^2-5t+4)(t^2-5t+6)dt$. (i) Find the critical points of the function $f\left(x\right)$. (ii) Find the points at which local minimum occurs. (iii) Find the points at which local maximum occurs. (iv) Find the number of zeros of the function $f\left(x\right)$ in $[0,5]$. [2020/20m/3a]

9. Find the maximum and the minimum value of the function $f\left(x\right)=2x^3-9x^2+12x+6$ on the interval $[2,3]$. [2019/15m/3a]

10. Find the maximum and the minimum values of $x^4-5x^2+4$ on the interval $[2,3]$. [2018/13m/4a]

11. For the cardioid $r=a(1+cos\theta )$, show that the square of the radius of curvature at any point $(r,\theta )$ is proportional to $r$. Also find the radius of curvature if $\theta =0,\frac{\pi }{4},\frac{\pi }{2}$. [2016/15m/8d]

12. A conical tent is of given capacity. For the least amount of Canvas required, for it, find the ratio of its height to the radius of its base. [2015/13m/2b]

13. Find the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $a$. [2014/15m/3a]

14. Let $p$ and $q$ be positive real numbers such that $\frac{1}{p}+\frac{1}{q}=1$. Show that for real numbers $a$, $b\ge 0$ $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ [2012/12m/1b]

2.3] Functions of two or three variables

32 questions

1. If $u(x,y,z)={(x^2+y^2+z^2)}^{-1/2}$, then find the value of $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}$. [2026/10m/1c]

2. If $u(x,y)=xf\left(\frac{y}{x}\right)+g\left(\frac{y}{x}\right)$, where $f$ and $g$ are arbitrary functions, then show that I. $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=xf\left(\frac{y}{x}\right),$ II\. $x^2\frac{{\partial }^{2}u}{\partial x^2}+2xy\frac{{\partial }^{2}u}{\partial x\partial y}+y^2\frac{{\partial }^{2}u}{\partial y^2}=0.$ [2025/10m/3c(ii)]

3. If $f(x,y)=\{\begin{array}{cc}\hfill xy\frac{x^2-y^2}{x^2+y^2},\hfill & \hfill \mathrm{\text{when}}(x,y)\ne (0,0)\hfill \\ \hfill 0,\hfill & \hfill \mathrm{\text{when}}(x,y)=(0,0),\hfill \end{array}$ then find $f_{xy}(0,0)\mathrm{\text{and}}{f}_{yx}(0,0).$ [2025/15m/4b]

4. If $u=(x+y)/(1-xy)$ and $v={tan}^{-1}x+{tan}^{-1}y$, then find $\partial (u,v)/\partial (x,y)$. Are $u$ and $v$ functionally related? If yes, find the relationship. [2024/15m/2b]

5. Justify whether $(0,0)$ is an extreme point for the function $f(x,y)=2x^4-3x^{2}y+y^2$. [2023/15m/3b]

6. A wire of length $l$ is cut into two parts which are bent in the form of a square and a circle respectively. Using Lagrange’s method of undetermined multipliers, find the least value of the sum of the areas so formed. [2022/15m/2b]

7. Given that $f(x,y)=|x^2-y^2|$. Find $f_{xy}(0,0)$ and $f_{yx}(0,0)$. Hence show that $f_{xy}(0,0)=f_{yx}(0,0)$. [2021/15m/2b]

8. If $u=x^2+y^2$, $v=x^2-y^2$, where $x=rcos\theta$, $y=rsin\theta$, then find $\frac{\partial (u,v)}{\partial (r,\theta )}$. [2021/7m/3a(i)]

9. Find an extreme value of the function $u=x^2+y^2+z^2$, subject to the condition $2x+3y+5z=30$, by using Lagrange’s method of undetermined multiplier. [2020/20m/4c]

10. Let $f:D(\subseteq {ℝ}^2)\to ℝ$ be a function and $(a,b)\in D$. If $f(x,y)$ is continuous at $(a,b)$, then show that the functions $f(x,b)$ and $f(a,y)$ are continuous at $x=a$ and at $y=b$ respectively. [2019/10m/1b]

11. If $u={sin}^{-1}\sqrt{\frac{x^{1/3}+y^{1/3}}{x^{1/2}+y^{1/2}}}$ then show that ${sin}^{2}u$ is a homogeneous function of $x$ and $y$ of degree $-\frac{1}{6}$. Hence show that $x^2\frac{{\partial }^{2}u}{\partial x^2}+2xy\frac{{\partial }^{2}u}{\partial x\partial y}+y^2\frac{{\partial }^{2}u}{\partial y^2}=\frac{tanu}{12}(\frac{13}{12}+\frac{{tan}^{2}u}{12})$ [2019/12m/4c(i)]

12. Using the Jacobian method, show that if $f\prime \left(x\right)=\frac{1}{1+x^2}$ and $f\left(0\right)=0$, then $f\left(x\right)+f\left(y\right)=f\left(\frac{x+y}{1-xy}\right)$ [2019/8m/4c(ii)]

13. Find the shortest distance from the point $(1,0)$ to the parabola $y^2=4x$. [2018/13m/2b]

14. Let $f(x,y)=\{\begin{array}{cc}\hfill x{y}^2,\hfill & \hfill \mathrm{\text{if}}y>0\hfill \\ \hfill -x{y}^2,\hfill & \hfill \mathrm{\text{if}}y\le 0\hfill \end{array}$ Determine which of $\frac{\partial f}{\partial x}(0,1)$ and $\frac{\partial f}{\partial y}(0,1)$ exists and which does not exist. [2018/12m/3b]

15. If $f(x,y)=\{\begin{array}{cc}\hfill \frac{xy(x^2-y^2)}{x^2+y^2},\hfill & \hfill (x,y)\ne (0,0)\hfill \\ \hfill 0,\hfill & \hfill (x,y)=(0,0),\hfill \end{array}$ calculate $\frac{{\partial }^{2}f}{\partial x\partial y}$ and $\frac{{\partial }^{2}f}{\partial y\partial x}$ at $(0,0)$. [2017/15m/3c]

16. Find the maximum and minimum values of $x^2+y^2+z^2$ subject to the conditions $\frac{x^2}{4}+\frac{y^2}{5}+\frac{z^2}{25}=1$ and $x+y-z=0$. [2016/20m/3a]

17. Let $f(x,y)=\{\begin{array}{cc}\hfill \frac{2x^{4}y-5x^2{y}^2+y^5}{{(x^2+y^2)}^2},\hfill & \hfill (x,y)\ne (0,0)\hfill \\ \hfill 0,\hfill & \hfill (x,y)=(0,0)\hfill \end{array}$ Find a $\delta >0$ such that $|f(x,y)-f(0,0)|<.01$, whenever $\sqrt{x^2+y^2}<\delta$. [2016/15m/3b]

18. Which point of the sphere $x^2+y^2+z^2=1$ is at the maximum distance from the point $(2,1,3)$? [2015/13m/3b]

19. For the function $f(x,y)=\{\begin{array}{cc}\hfill \frac{x^2-x\sqrt{y}}{x^2+y},\hfill & \hfill (x,y)\ne (0,0)\hfill \\ \hfill 0\hfill & \hfill (x,y)=(0,0)\hfill \end{array}$ Examine the continuity and differentiability. [2015/12m/4d]

20. Find the maximum or minimum values of $x^2+y^2+z^2$ subject to the conditions $a{x}^2+b{y}^2+c{z}^2=1$ and $lx+my+nz=0$. Interpret the result geometrically. [2014/20m/3b]

21. Using Lagrange’s multiplier method, find the shortest distance between the line $y=10-2x$ and the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$. [2013/20m/3a]

22. Compute $f_{xy}(0,0)$ and $f_{yx}(0,0)$ for the function $f(x,y)=\{\begin{array}{cc}\hfill \frac{x{y}^3}{x+y^2},\hfill & \hfill (x,y)\ne (0,0)\hfill \\ \hfill 0,\hfill & \hfill (x,y)=(0,0).\hfill \end{array}$ Also, discuss the continuity of $f_{xy}$ and $f_{yx}$ at $(0,0)$. [2013/15m/3b]

23. Define a function $f$ of two real variables in the $xy$-plane by $f(x,y)=\{\begin{array}{cc}\hfill \frac{x^{3}cos\frac{1}{y}+y^{3}cos\frac{1}{x}}{x^2+y^2}\hfill & \hfill \mathrm{\text{for}}x,y\ne 0\hfill \\ \hfill 0,\hfill & \hfill \mathrm{\text{otherwise}}\hfill \end{array}$ Check the continuity and differentiability of $f$ at $(0,0)$. [2012/12m/1a]

24. Find the points of local extrema and saddle points of the function $f$ of two variables defined by $f(x,y)=x^3+y^3-63(x+y)+12xy$ [2012/20m/3a]

25. Find ${lim}_{(x,y)\to (0,0)}\frac{x^{2}y}{x^3+y^3}$ if it exists. [2011/10m/1c]

26. Find the points on the sphere $x^2+y^2+z^2=4$ that are closest to and farthest from the point $(3,1,-1)$. [2011/20m/3b]

27. Show that a box (rectangular parallelopiped) of maximum volume $V$ with prescribed surface area is a cube. [2010/20m/2b]

28. If $f(x,y)$ is a homogeneous function of degree $n$ in $x$ and $y$, and has continuous first- and second-order partial derivatives, then show that (i) $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf$ \(ii\) $x^2\frac{{\partial }^{2}f}{\partial x^2}+2xy\frac{{\partial }^{2}f}{\partial x\partial y}+y^2\frac{{\partial }^{2}f}{\partial y^2}=n(n-1)f$ [2010/20m/4b]

29. If $x=3\pm 0.01$ and $y=4\pm 0.01$, with approximately what accuracy can you calculate the polar coordinates $r$ and $\theta$ of the point $P(x,y)$? Express your estimates as percentage changes of the values that $r$ and $\theta$ have at the point $(3,4)$. [2009/20m/2b]

30. Let $f:{ℝ}^2\to ℝ$ be defined as $f(x,y)=\{\begin{array}{cc}\hfill \frac{xy}{\sqrt{x^2+y^2}},\hfill & \hfill \mathrm{\text{if}}(x,y)\ne (0,0)\hfill \\ \hfill 0,\hfill & \hfill \mathrm{\text{if}}(x,y)=(0,0)\hfill \end{array}$ Is $f$ continuous at $(0,0)$? Compute partial derivatives of $f$ at any point $(x,y)$, if exist. [2009/20m/3b]

31. A space probe in the shape of the ellipsoid $4x^2+y^2+4z^2=16$ enters the earth’s atmosphere and its surface begins to heat. After one hour, the temperature at the point $(x,y,z)$ on the probe surface is given by $T(x,y,z)=8x^2+4yz-16z+600$ Find the hottest point on the probe surface. [2009/20m/3c]

32. Determine the maximum and minimum distances of the origin from the curve given by the equation $3x^2+4xy+6y^2=140.$ [2008/20m/2b]

2.4] Integration

18 questions

1. Examine the convergence of the integral ${\int }_0^1\frac{logx}{1+x}dx$ [2023/10m/1d]

2. Examine the convergence of ${\int }_0^2\frac{dx}{(2x-x^2)}$. [2022/10m/1d]

3. If ${\int }_0^{x}f\left(t\right)dt=x+{\int }_x^{1}tf\left(t\right)dt$, then find the value of $f\left(1\right)$. [2021/5m/3a(ii)]

4. Express ${\int }_a^b{(x-a)}^m{(b-x)}^{n}dx$ in terms of Beta function. [2021/8m/3a(iii)]

5. Evaluate ${\int }_0^1{tan}^{-1}(1-\frac{1}{x})dx$. [2020/15m/2a]

6. Find the limit ${lim}_{n\to \infty }\frac{1}{n^2}{\sum }_{r=0}^{n-1}\sqrt{n^2-r^2}$. [2018/10m/1d]

7. Examine if the improper integral ${\int }_0^3\frac{2xdx}{{(1-x^2)}^{2/3}}$ exists. [2017/10m/4c]

8. Evaluate : $I={\int }_0^1\sqrt[3]{xlog\left(\frac{1}{x}\right)}dx$ [2016/10m/1c]

9. Evaluate the following integral : ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{\sqrt[3]{sinx}}{\sqrt[3]{sinx}+\sqrt[3]{cosx}}dx.$ [2015/10m/1d]

10. Evaluate : ${\int }_0^1\frac{{log}_e(1+x)}{1+x^2}dx$ [2014/10m/1d]

11. Evaluate ${\int }_0^1(2xsin\frac{1}{x}-cos\frac{1}{x})dx$. [2013/10m/1c]

12. Define a sequence $s_n$ of real numbers by $s_n={\sum }_{i=1}^n\frac{{(log(n+i)-logn)}^2}{n+i}$ Does ${lim}_{n\to \infty }{s}_n$ exist? If so, compute the value of this limit and justify your answer. [2012/20m/3b]

13. Find all the real values of $p$ and $q$ so that the integral ${\int }_0^1{x}^p{(log\frac{1}{x})}^{q}dx$ converges. [2012/20m/3c]

14. ${\int }_0^{1}lnxdx$. [2011/12m/3a(ii)]

15. Does the integral ${\int }_{-1}^1\sqrt{\frac{1+x}{1-x}}dx$ exist? If so, find its value. [2010/12m/1d]

16. Show that the function $f\left(x\right)=\left[x^2\right]+|x-1|$ is Riemann integrable in the interval $[0,2]$, where $[\alpha ]$ denotes the greatest integer less than or equal to $\alpha$. Can you give an example of a function that is not Riemann integrable on $[0,2]$? Compute ${\int }_0^{2}f\left(x\right)dx$, where $f\left(x\right)$ is as above. [2010/12m/1f]

17. If $f$ is the derivative of some function defined on $[a,b]$, prove that there exists a number $\eta \in [a,b]$, such that ${\int }_a^{b}f\left(t\right)dt=f\left(\eta \right)(b-a)$ [2009/12m/1d]

18. Evaluate ${\int }_0^1{(xlnx)}^{3}dx$. [2008/12m/1d]

2.5] Multiple integrals, areas and volumes

23 questions

1. Evaluate the double integral ${\int }_0^2{\int }_{x^2+1}^{2x+1}{x}^{2}ydydx$ by reversing the order of integration. [2026/10m/3b(i)]

2. Find the volume of the solid, which is formed from the intersection of the coordinate planes $x=0$, $y=0$, $z=0$ and the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$. [2026/10m/3b(ii)]

3. Evaluate ${\iint }_R^{}ydxdy$, where $R$ is the region bounded by $y=x$ and $y=4x-x^2$. [2025/10m/3c(i)]

4. Using double integration, find the area lying inside the cardioid $r=a(1+cos\theta )$ and outside the circle $r=a$. [2024/15m/4b]

5. Evaluate the triple integral which gives the volume of the solid enclosed between the two paraboloids $Z=5(x^2+y^2)$ and $Z=6-7x^2-y^2$. [2023/15m/2b]

6. Use double integration to calculate the area common to the circle $x^2+y^2=4$ and the parabola $y^2=3x$. [2022/15m/3b]

7. Show that the entire area of the Astroid : $x^{2/3}+y^{2/3}=a^{2/3}$ is $\frac{3}{8}\pi a^2$. [2021/15m/4b]

8. The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ revolves about the $x$-axis. Find the volume of the solid of revolution. [2018/13m/2c]

9. Evaluate the integral ${\int }_0^a{\int }_{x/a}^x\frac{xdydx}{x^2+y^2}$. [2018/12m/4b]

10. Integrate the function $f(x,y)=xy(x^2+y^2)$ over the domain $R:\{-3\le x^2-y^2\le 3,1\le xy\le 4\}$. [2017/10m/1c]

11. Find the volume of the solid above the $xy$-plane and directly below the portion of the elliptic paraboloid $x^2+\frac{y^2}{4}=z$ which is cut off by the plane $z=9$. [2017/15m/2a]

12. Prove that $\frac{\pi }{3}\le {\iint }_D^{}\frac{dxdy}{\sqrt{x^2+{(y-2)}^2}}\le \pi$ where $D$ is the unit disc. [2017/10m/4d]

13. Find the surface area of the plane $x+2y+2z=12$ cut off by $x=0$, $y=0$ and $x^2+y^2=16$. [2016/15m/3c]

14. Evaluate ${\iint }_R^{}f(x,y)dxdy$ over the rectangle $R=[0,1;0,1]$ where $f(x,y)=\{\begin{array}{cc}\hfill x+y,\hfill & \hfill \mathrm{\text{if}}{x}^2<y<2x^2\hfill \\ \hfill 0,\hfill & \hfill \mathrm{\text{elsewhere}}\hfill \end{array}$ [2016/15m/4c]

15. Evaluate the integral ${\iint }_R^{}{(x-y)}^2{cos}^2(x+y)dxdy$ where $R$ is the rhombus with successive vertices as $(\pi ,0)$ $(2\pi ,\pi )$ $(\pi ,2\pi )$ $(0,\pi )$. [2015/12m/3d]

16. Evaluate ${\iint }_R^{}\sqrt{|y-x^2|}dxdy$ where $R=[-1,1;0,2]$. [2015/13m/4a]

17. By using the transformation $x+y=u$, $y=uv$, evaluate the integral $\iint \{xy(1-x-y){\}}^{1/2}dxdy$ taken over the area enclosed by the straight lines $x=0$, $y=0$ and $x+y=1$. [2014/15m/2c]

18. Evaluate ${\iint }_D^{}xydA$, where $D$ is the region bounded by the line $y=x-1$ and the parabola $y^2=2x+6$. [2013/15m/3c]

19. Compute the volume of the solid enclosed between the surfaces $x^2+y^2=9$ and $x^2+z^2=9$. [2012/20m/4a]

20. Find the volume of the solid that lies under the paraboloid $z=x^2+y^2$ above the $xy$-plane and inside the cylinder $x^2+y^2=2x$. [2011/20m/3c]

21. Let $D$ be the region determined by the inequalities $x>0$, $y>0$, $z<8$ and $z>x^2+y^2$. Compute ${\iiint }_D^{}2xdxdydz$ [2010/20m/3b]

22. Evaluate the double integral ${\int }_0^a{\int }_y^a\frac{xdxdy}{x^2+y^2}$ by changing the order of integration. [2008/20m/3b]

23. Obtain the volume bounded by the elliptic paraboloide given by the equations $z=x^2+9y^2\mathrm{\text{and}}z=18-x^2-9y^2.$ [2008/20m/4b]

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Last updated: 30 October 2026