# Calculus

 Question 1
The value of the integral $\iint_{R} x y d x d y$ over the region $R$, given in the figure, is ___
(rounded off to the nearest integer).

 A 0 B 1 C 2 D 3
GATE EC 2023   Engineering Mathematics
Question 1 Explanation:

\begin{aligned} I & =\iint_{R} x y d x d y \\ & =\int_{y=0}^{1} \int_{x=-y}^{y} x y d x d y+\int_{y=1}^{2} \int_{x=y-2}^{2-y} x y d x d y \\ & =\int_{0}^{1} y\left(\frac{x^{2}}{2}\right)_{-y}^{y} d y+\int_{1}^{2} y\left(\frac{x^{2}}{2}\right)_{y-2}^{2-y} d y \\ & =0+0=0 \end{aligned}
 Question 2
The value of the line integral $\int_{P}^{Q}\left(z^{2} d x+3 y^{2} d y+2 x z d z\right)$ along the straight line joining the points $P(1,1,2)$ and $Q(2,3,1)$ is
 A 20 B 24 C 29 D -5
GATE EC 2023   Engineering Mathematics
Question 2 Explanation:
$\int_{P}^{Q} z^{2} d x+3 y^{2} d y+2 x z d z$ along the line joining the points $P(1,1,2)$ and $Q(2,3,1)$ is
\begin{aligned} & =\int_{P(1,2)}^{P(2,1)} z^{2} d x+2 x y d z+\int_{y=1}^{3} 3 y^{2} d y \\ & =\left(x z^{2}\right)_{(1,2)}^{(2,1)}+\left(y^{3}\right)_{1}^{3} \\ & =\left(2 \times 1^{2}-1 \times 2^{2}\right)+\left(3^{3}-1^{3}\right) \\ & =-2+26=24 \end{aligned}

 Question 3
The value of the contour integral, $\oint_{c}\left(\frac{z+2}{z^{2}+2 z+2}\right) d z$, where the contour $C$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$, taken in the counter clockwise direction, is
 A $-\pi(1+j)$ B $\pi(1+j)$ C $\pi(1-j)$ D $-\pi(1-j)$
GATE EC 2023   Engineering Mathematics
Question 3 Explanation:
$I=\oint_{c} \frac{z+2}{z^{2}+2 z+2} d z ; \quad c=\left|z+1-\frac{3}{2} i\right|=1$

Poles are given $(z+1)^{2}+1=0$
\begin{aligned} & z+1= \pm \sqrt{-1} \\ & z=-1+j,-1-j \end{aligned}
where $-1-i$ lies outside ' $c$ '
$z=(-1,1) \text { lies inside } 'c'$.

by $\mathrm{CRT}$
\begin{aligned} \oint_{c} f(z) d z & =2 \pi i \operatorname{Res}(f(z), z=-1+j) \\ & =2 \pi i\left(\frac{z+2}{2(z+1)}\right)_{z=-1+i}\\ & =2 \pi i\left(\frac{-1+j+2}{2(-1+j+1)}\right) \\ & =\pi(1+j) \end{aligned}
 Question 4
The value of the integral
$\int \int _D 3(x^2+y^2)dxdy$
, where $D$ is the shaded triangular region shown in the diagram, is _____ (rounded off to the nearest integer).

 A 128 B 1024 C 512 D 64
GATE EC 2022   Engineering Mathematics
Question 4 Explanation:
\begin{aligned} I&=\int_{0}^{4}\int_{-x}^{x}(3x^2+3y^2)dydx\\ &=\int_{0}^{4}\left [ 3x^y+\frac{3y^3}{3} \right ]_{-x}^xdx\\ &=\int_{0}^{4}[3x^2(2x)+2x^3]dx\\ &=\int_{0}^{4}8x^3 dx\\ &=2 \times 4^4\\ &=512 \end{aligned}
 Question 5
The function $f(x)=8 \log _e x-x^2+3$ attains its minimum over the interval $[1,e]$ at $x=$ _________.
(Here $\log _e x$ is the natural logarithm of $x=$.)
 A 2 B 1 C $e$ D $\frac{1+e}{2}$
GATE EC 2022   Engineering Mathematics
Question 5 Explanation:
$f(x)=8 \log _e x-x^2+3$ when $x \in [1,e]$
Differentiating both side,
\begin{aligned} f(x)&=\frac{8}{x}-2x=0 \; where\; x \gt 0\\ f'(x)&=0\\ \frac{8}{x}-2x&=0\\ 8-2x^2&=0\\ x^2&=4\\ x&=\pm 2 \end{aligned}
\begin{aligned} f''(x)&=\frac{-8}{x^2}-2 \\ f''(2)&=-6 \lt 0\\ \end{aligned}
f(x) is maximum for x = 2
Minimum of f(x) will be in [1, e] = min [f(1), f(e)]
$f(e)=8 \ln e -e^2+3=3.61$
Hence, minimum value of f(x) occurs at x=1

There are 5 questions to complete.