calculus

Domain A is bounded by curve x² = 4y, ordinate x = 2, and x axis. The value of ∫∫A y dxdy is

The exact solution of $\int_0^4 \frac{dx}{1+x}$ is represented as n. If m represents numerically evaluated value of the above integral using Trapezoidal rule by considering four eq...

The values of a function f obtained for different values of x are shown in the table below. Using Simpson's one-third rule, $\int_0^1 f(x) dx \approx$ ________ (rounded off to 2 de...

In the closed interval [0,3], the minimum value of the function f given below is f(x) = 2x³ - 9x² + 12x

$$ \text { The values of a function } f \text { obtained for different values of } x \text { are shown in the table below. } $$ $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 0.25...

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

Let f(.) be a twice differentiable function from R² → R. If p, x₀ ∈ R² where ||p|| is sufficiently small (here ||. || is the Euclidean norm or distance function), then f(x₀ + p) =...

If the value of the double integral $\int_{x=3}^{4}\int_{y=1}^{2}\frac{dydx}{(x+y)^2}$ is $\log_e (a/24)$, then $a$ is ________ (answer in integer).

If the value of the double integral $\int_{x=3}^{4} \int_{y=1}^{2} \frac{dydx}{(x + y)^2}$ is $\log_e(\frac{a}{24})$, then $a$ is __________ (answer in integer).

A linear transformation maps a point (x, y) in the plane to the point (x̂, ŷ) according to the rule x̂ = 3y, ŷ = 2x. Then, the disc x² + y² ≤ 1 gets transformed to a region with an...

The smallest perimeter that a rectangle with area of 4 square units can have is ______ units. (Answer in integer)

The limit $\rm p = \displaystyle\lim_{x \rightarrow \pi} \left( \frac{x^2 + α x + 2 \pi^2}{x - \pi + 2 \sin x} \right)$ has a finite value for a real α . The value of α and the cor...

The Fourier series expansion of x 3 in the interval −1 ≤ x < 1 with periodic continuation has

If $f(x)=2\ln(\sqrt{e^x})$ , what is the area bounded by f(x) for the interval [0, 2] on the x-axis?

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

A polynomial ψ(s) = a n s n + a n-1 s n-1 + ......+ a 1 s + a 0 of degree n > 3 with constant real coefficients a n , a n-1 , ... a 0 has triple roots at s = -σ. Which one of the f...

If y(x) satisfies the differential equation (sin x) dy/dx + y cos x = 1, subject to the condition y(π/2) = π/2, then y(π/6) is

The value of lim (x→0) (1-cos x)/x^2 is

Value of $\int_{4}^{5.2} \ln x \, dx$ using Simpson's one-third rule with interval size 0.3 is

The value of $\int_0^{\pi/2} \int_0^{\cos\theta} r \sin\theta \, dr \, d\theta$ is

Let $f(x) = x^2 - 2x + 2$ be a continuous function defined on $x \in [1, 3]$. The point $x$ at which the tangent of $f(x)$ becomes parallel to the straight line joining $f(1)$ and...

Define [x] as the greatest integer less than or equal to x, for each x ϵ (-∞, ∞). If y = [x], then area under y for x ϵ [1,4] is

A parabola $x = y^2$ with $0 \le x \le 1$ is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by $360^\circ$ around the x-axis is

The value of the following definite integral is _________ (round off to three decimal places) $\int_{1}^{e} (x \ln x) dx$

The divergence of the vector field $\vec{u} = e^x (\cos y\hat{i} + \sin y \hat{j})$ is

The value of lim_{x\to 0} \frac{x^3 - \sin(x)}{x} is

According to the Mean Value Theorem, for a continuous function $f(x)$ in the interval $[a,b]$, there exists a value $\xi$ in this interval such that $\int_{a}^{b} f(x)dx =$

The Laplace transform of $te^t$ is

For a loaded cantilever beam of uniform cross-section, the bending moment (in N·mm) along the length is $M(x) = 5x^2 + 10x$, where x is the distance (in mm) measured from the free...

The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is

A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\,\,$$ in the range $$0 \le u \le 1$$ is rotated about the $$...

$$\mathop {Lt}\limits_{x \to \infty } \left( {\sqrt {{x^2} + x - 1} - x} \right)$$ is

The values of $$x$$ for which the function $$f\left( x \right) = {{{x^2} - 3x - 4} \over {{x^2} + 3x - 4}}$$ is NOT continuous are

$$\mathop {Lt}\limits_{x \to 0} {{{{\log }_e}\left( {1 + 4x} \right)} \over {{e^{3x}} - 1}}$$ is equal to

Consider the function $$f\left( x \right) = 2{x^3} - 3{x^2}\,\,$$ in the domain $$\,\left[ { - 1,2} \right].$$ The global minimum of $$f(x)$$ is _________.

Using a unit step size, the value of integral $$\int\limits_1^2 {x\,\ln \,xdx\,\,\,} $$ by trapezoidal rule is ___________.

The value of $$\mathop {Lim}\limits_{x \to 0} \,{{1 - \cos \left( {{x^2}} \right)} \over {2{x^4}}}$$ is

The value of $$\mathop {Lim}\limits_{x \to 0} \left( {{{ - \sin x} \over {2\sin x + x\cos x}}} \right)\,\,\,$$ is __________.

At $$x=0,$$ the function $$f\left( x \right) = \left| x \right|$$ has

The value of the integral $$\,\int\limits_0^2 {\int\limits_0^x {{e^{x + y}}\,\,dy} } $$ $$dx$$ is

$$\mathop {Lt}\limits_{x \to 0} {{x - \sin x} \over {1 - \cos x}}$$ is

If a function is continuous at a point,

$$\mathop {Lt}\limits_{x \to 0} \left( {{{{e^{2x}} - 1} \over {\sin \left( {4x} \right)}}} \right)\,\,$$ is equal to

If $$\,y = f\left( x \right)\,\,$$ is the solution of $${{{d^2}y} \over {d{x^2}}} = 0$$ with the boundary conditions $$y=5$$ at $$x=0,$$ and $$\,{{dy} \over {dx}} = 2$$ at $$x=10,$...

The value of the definite integral $$\int_1^e {\sqrt x \ln \left( x \right)dx} $$ is

$$\,\mathop {Lim}\limits_{x \to 0} \left( {{{1 - \cos x} \over {{x^2}}}} \right)$$ is

At $$x=0,$$ the function $$f\left( x \right) = {x^3} + 1$$ has

Consider the function $$f\left( x \right) = \left| x \right|$$ in the interval $$\,\, - 1 \le x \le 1.\,\,\,$$ At the point $$x=0, f(x)$$ is

A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profile of the arch follows the equation $$y = 2x\,\,\, - 0.1{x^2}...

The area enclosed between the straight line $$y=x$$ and the parabola $$y = {x^2}$$ in the $$x-y$$ plane is

A series expansion for the function $$\sin \theta $$ is _______.

What is $$\mathop {Lim}\limits_{\theta \to 0} {{\sin \theta } \over \theta }\,\,$$ equal to ?

If $$f(x)$$ is even function and a is a positive real number , then $$\int\limits_{ - a}^a {f\left( x \right)dx\,\,} $$ equals ________.

The function $$y = \left| {2 - 3x} \right|$$

The parabolic are $$y = \sqrt x ,1 \le x \le 2$$ is revolved around the $$x$$-axis. The volume of the solid of revolution is

The value of the integral $$\int\limits_{ - a}^a {{{dx} \over {1 + {x^2}}}} $$

The area enclosed between the curves $${y^2} = 4x\,\,$$ and $${{x^2} = 4y}$$ is

The distance between the origin and the point nearest to it on the surface $$\,\,{z^2} = 1 + xy\,\,$$ is

Let $$\,\,f = {y^x}.$$ What is $$\,\,{{{\partial ^2}f} \over {\partial x\partial y}}\,\,$$ at $$x=2,$$ $$y=1$$?

The value of $$\,\,\mathop {Lim}\limits_{x \to 8} {{{x^{1/3}} - 2} \over {x - 8}}\,\,$$ is

Which of the following integrals is unbounded?

The length of the curve $$\,y = {2 \over 3}{x^{3/2}}$$ between $$x=0$$ & $$x=1$$ is

In the Taylor series expansion of $${e^x}$$ about $$x=2,$$ the coefficient of $$\,\,{\left( {x - 2} \right)^4}\,\,$$ is

If $$\,\,\,y = x + \sqrt {x + \sqrt {x + \sqrt {x + .....\alpha } } } \,\,\,$$ then $$y(2)=$$ __________.

$$\mathop {Lim}\limits_{x \to 0} {{{e^x} - \left( {1 + x + {{{x^2}} \over 2}} \right)} \over {{x^3}}} = $$

The minimum value of function $$\,\,y = {x^2}\,\,$$ in the interval $$\,\,\left[ {1,5} \right]\,\,$$ is

By a change of variables $$x(u, v) = uv,$$ $$\,\,y\left( {u,v} \right) = {v \over u}$$ in a double integral, the integral $$f(x, y)$$ changes to $$\,\,\,f\left( {uv,{\raise0.5ex\hb...

$$\int\limits_{ - a}^a {\left[ {{{\sin }^6}\,x + {{\sin }^7}\,x} \right]dx} $$ is equal to

Changing the order of integration in the double integral $${\rm I} = \int\limits_0^8 {\int\limits_{{\raise0.5ex\hbox{$\scriptstyle x$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\...

If $$\,\,\,x = a\left( {\theta + Sin\theta } \right)$$ and $$y = a\left( {1 - Cos\theta } \right)$$ then $$\,\,{{dy} \over {dx}} = \,\_\_\_\_\_.$$

The volume of an object expressed in spherical co-ordinates is given by $$V = \int\limits_0^{2\pi } {\int\limits_0^{{\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em...

Value of the function $$\mathop {Lim}\limits_{x \to a} \,{\left( {x - a} \right)^{x - a}}$$ is _______.

Area bounded by the curve $$y = {x^2}$$ and the lines $$x=4$$ and $$y=0$$ is given by

If a function is continuous at a point its first derivative

The area bounded by the parabola $$2y = {x^2}$$ and the lines $$x=y-4$$ is equal to _________.

If $$H(x, y)$$ is homogeneous function of degree $$n$$ then $$x{{\partial H} \over {\partial x}} + y{{\partial H} \over {\partial y}} = nH$$

The function $$f\left( {x,y} \right) = {x^2}y - 3xy + 2y + x$$ has

$$\mathop {Lim}\limits_{x \to 0} {{x\left( {{e^x} - 1} \right) + 2\left( {\cos x - 1} \right)} \over {x\left( {1 - \cos x} \right)}} = \_\_\_\_\_\_.$$