Recent questions tagged calculus

Domain $A$ is bounded by curve $x^{2}=4 y$, ordinate $x=2$, and $x$ axis.The value of $\iint_{A} y d x d y$ is$1 / 5$$1 / 3$$5 / 12$$1 / 2$

The exact solution of $\int_{0}^{4} \frac{d x}{1+x}$ is represented as $n$.If $m$ represents numerically evaluated value of the above integral using Trapezoidal rule by c...

For the differential equation given below, which one of the following options is correct?$$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 ...

The values of a function $f$ obtained for different values of $x$ are shown in the table below.$x$00.250.50.751.0$f(x)$0.92.01.51.80.4Using Simpson's one-third rule,$\int...

​​In the closed interval $[0,3]$, the minimum value of the function $f$ given below is$$ f(x)=2 x^{3}-9 x^{2}+12 x$$ $0$$4$$5$$9$

The directional derivative of the function $f$ given below at the point $(1,0)$ in the direction of $\frac{1}{2}(\hat{\imath}+\sqrt{3} \hat{\jmath})$ is __________ (round...

Let $y$ be the solution of the differential equation with the initial conditions given below. If $y(x=2)=A \ln 2$, then the value of $A$ is ________ (rounded off to $2$ d...

​​The value of the surface integralwhere $S$ is the external surface of the sphere $x^{2}+y^{2}+z^{2}=R^{2}$ is$0$$4 \pi R^{3}$$\frac{4 \pi}{3} R^{3}$$\pi R^{3}$

​​​​​Let $f(z)$ be an analytic function, where $z=x+i y$. If the real part of $f(z)$ is $\cosh x \cos y$, and the imaginary part of $f(z)$ is zero for $y=0$, then $f(z)$ ...

​​​Let $f(.)$ be a twice differentiable function from $\mathbb{R}^2 \rightarrow \mathbb{R}$. If $\boldsymbol{p}, x_0 \in \mathbb{R}^2$ where $\|p\|$ is sufficiently small...

​​​​The velocity field of a two-dimensional, incompressible flow is given by\[\vec{V}=2 \sinh x \hat{\imath}+v(x, y) \hat{\jmath}\]where $\hat{\imath}$ and $\hat{\jmath}$...

​​​​A plane, solid slab of thickness $L$, shown in the figure, has thermal conductivity $k$ that varies with the spatial coordinate $x$ as $k=A+B x$, where $A$ and $B$ ar...

If the value of the double integral\[\int_{x=3}^{4} \int_{y=1}^{2} \frac{d y d x}{(x+y)^{2}}\]is $\log _{e}(a / 24)$, then $a$ is __________ (answer in integer).

Consider a slab of $20 \mathrm{~mm}$ thickness. There is a uniform heat generation of $\dot{q}=100 \mathrm{MW} / \mathrm{m}^{3}$ inside the slab. The left and right faces...

The figure shows the plot of a function over the interval $[-4,4]$. Which one of the options given $\text{CORRECTLY}$$ identifies the function?$|2-x|$$|2-| x||$$|2+| x||$...

Consider a laterally insulated rod of length $\mathrm{L}$ and constant thermal conductivity. Assuming one-dimensional heat conduction in the rod, which of the following s...

The value of $k$ that makes the complex-valued function$$f(z)=e^{-k x}(\cos 2 y-i \sin 2 y)$$analytic, where $z=x+i y$, is __________. (Answer in integer)

An explosion at time $t=0$ releases energy $\text{E}$ at the origin in a space filled with a gas of density $\rho$. Subsequently, a hemispherical blast wave propagates ra...

Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point $(1, 2, 3)$. The surface integral $\int _{A} \vec{F}.d\vec{A}$ ...

Consider the definite integral$$\int_{1}^{2} \left ( 4x^{2} + 2x + 6 \right )dx.$$Let $I_{e}$ be the exact value of the integral. If the same integral is estimated using ...

Given $\int_{-\infty }^{\infty } e^{-x^{2}} dx = \sqrt{\pi }.$If $a$ and $b$ are positive integers, the value of $\int_{-\infty }^{\infty } e^{-a\left ( x+b \right )^{2}}...

A polynomial $\varphi \left ( s \right ) = a_{n}s^{n} + a_{n-1}s^{n-1} + \cdots + a_{1}s+a_{0}$ of degree $n>3$ with constant real coefficients $a_{n}, a_{n-1}, \:\dots a...

Given $z = x + iy, i = \sqrt{-1}$. $C$ is a circle of radius $2$ with the centre at the origin. If the contour $C$ is traversed anticlockwise, then the value of the integ...

​​​​​​The limit $$p = \lim_{x\rightarrow \pi} \left ( \frac{x^{2} + \alpha x + 2 \pi^{2}}{x - \pi + 2 \sin x } \right )$$ has a finite value for a real $\alpha$. The valu...

Given a function $\varphi =\frac{1}{2}\left (x ^{2} + y^{2} + z^{2} \right )$ in three-dimensional Cartesian space, the value of the surface integral $$\oint_{s}\hat{n}\...

The Fourier series expansion of $x^{3}$ in the interval $-1 \leq x < 1$ with periodic continuation hasonly sine termsonly cosine termsboth sine and cosine termsonly sine ...

A tiny temperature probe is fully immersed in a flowing fluid and is moving with zero relative velocity with respect to the fluid. The velocity field in the fluid is $\ve...

The value of the integral $$\oint \left ( \frac{6z}{2z^{4} - 3z^{3} + 7z^{2} - 3z + 5 } \right )dz$$ evaluated over a counter-clockwise circular contour in the complex pl...

A two dimensional flow has velocities in $x$ and $y$ directions given by $u = 2xyt$ and $v = -y^{2}t$, where $\text{t}$ denotes time. The equation for streamline passing ...