Recent questions in Calculus

Let $L$ be the lamina of the form $x^{2}+4 y^{2} \leq 64,0 \leq y \leq 4$, with density $\rho(x, y)=\mid x \mid y$. Then, the mass of $L$ (in integer) is $\_\_\_\_$

Match each entry of $\text{List-1}$ with a suitable entry in $\text{List-2}$ and choose the correct option.$\textbf{List-1}$$\textbf{List-2}$$\mathrm{P}$$\text{The sum of...

​Consider the infinite series$$ \begin{array}{l} \text { (P) }: \sum_{n=2}^{\infty} \frac{1}{(n \log n)^{1 / n}} \\\text { (Q): } \sum_{n=1}^{\infty} \frac{n^{n}}{(2 n)!}...

​​Consider the function$$ f(x, y)=x^{2} y+2 x y^{2}-2 x^{2} y^{2}.$$Then which one of the following statements is correct?$\left(\frac{3}{2}, 0\right)$ is a point of loca...

Consider, $i$ and $j$ are unit vectors along $x$ and $y$ directions of a Cartesian $(x, y)$ coordinate system, respectively and $t$ is time. Temperature ($T$) and fluid v...

Let\[f(x)=\left\{\begin{array}{cr}\pi+x, & -\pi \leq x<0, \\0, & 0 \leq x<\pi,\end{array}\right.\]with $f(x+2 \pi)=f(x)$. If $F(x)$ represents the Fourier series of $f(x)...

Assume that $f:[0,1] \rightarrow \mathbb{R}$ is continuous on $[0,1]$ and differentiable on $(0,1)$ such that $f(x+h)=f(x)+h f^{\prime}(x+\theta h)$ for some $0<\theta<1$...

If\[\int_{0}^{\alpha} \int_{\sqrt{x / \alpha}}^{1} e^{y^{3}} d y d x=e-1, \quad \alpha>0,\]then the value $\text{(in integer)}$ of $\alpha$ is __________.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined by $f(x, y)=\left\{\begin{array}{cc}\frac{x y}{|x|+y}, & y \neq-|x| \\ 0, & \text { otherwise. }\end{...

If the polynomial $P(x)=a_{0}+a_{1} x+a_{2} x(x-1)+a_{3} x(x-1)(x-2)$interpolates the points $(0,2),(1,3),(2,2)$, and $(3,5)$, then the value of $P\left(\frac{5}{2}\right...

The surface area of the portion of the paraboloid $z=x^{2}+y^{2}$that lies between the planes $z=0$ and $z=\frac{1}{4}$ is$\frac{\pi}{6}(2 \sqrt{2}-1)$$\frac{\pi}{2}(2 \s...

Which one of the given figures $P, Q, R$ and $S$ represents the graph of the following function?$$f(x)=|| x+2|-| x-1||$$$\text{P}$$\text{Q}$$\text{R}$$\text{S}$

The value of$\lim _{x \rightarrow 0} \frac{1}{x} \int_{2}^{2+x}\left(t+\sqrt{t^{2}+5}\right) d t$$0$$4$$5$$6$

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be the continuous function such that $f(x)=2+\frac{g(x)}{x}$ for all $x>0$, where $g(x)=\int_{1}^{x} f(t) d t$ for all $x>0$. T...

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by $f(x, y)=4 x y-2 x^{2}-y^{4}+1$. The number of critical points where $f$ has local maximum is equal to