Curve Tracing

The area bounded by the parabolas  $y={(x+1)}^2$  and  $y={(x-1)}^2$  and the line y = 1/4 is:

Find the period of the following function: $y=\sin \left(4x+3\right)$.

Find the period of the following function: $y=\sin \left(2x+3\right)$.

Check whether the following curve is symmetric about line $y=x$ or not.

$x^2-y^2=2xy$

Check whether the following curve is symmetric about line $y=x$ or not.

$x^2+y^2=2xy$

Check whether the following curve is symmetric about line $y=x$ or not.

$x^2-y^2=1$

Check whether the following curve is symmetric about line $y=x$ or not.

$x^2+y^2=1$

Sketch the curve $r={\sec }^2\theta$ for the interval $0⩽\theta ⩽2\pi .$

Sketch the curve $r=1-2\sin \theta$ for the interval $\left[0, 2\pi \right]$. Sketch also the inner loop and state its domain.

A curve, $C,$ has polar equation $r=2-\sin \theta ,$ for $0⩽\theta ⩽2\pi .$ Sketch $C.$

Convert the following polar coordinates to its equivalent Cartesian coordinates. $(2,\pi )$

Area bounded by the curve $y=\left|\left|x\right|-2\right|$ and the line $y=2,$ is

Consider the following parametric equation of a curve:

$x\left(\theta \right)=\left|\cos 4\mathrm{\theta }\right|\mathrm{cos\theta }$

$y\left(\theta \right)=\left|\cos 4\mathrm{\theta }\right|\sin \theta$

For $0\le \mathrm{\theta }\le 2\mathrm{\pi }$

Which one of the following graphs represents the curve?

For a real numbers $x$, let $\left[x\right]$ denote the largest number less than or equal $x.$For $x\in R$ let $f\left(x\right)=\left[x\right]\sin \pi x$ .Then

The area of the closed figure bounded by $x=-1,\mathrm{ }x=2$ and $y=\left\{\begin{array}{cc}-x^2+2,& x\le 1\\ 2x-1,& x>1\end{array}\right.$ and the $x$-axis is

Area enclosed by the curve $y=f\left(x\right)$ defined parametrically as $x=\frac{1-t^2}{1+t^2},y=\frac{2t}{1+t^2}$ is equal to (where $t$ is parameter)

Area bounded by the curves $\mathrm{y}=\left|\mathrm{ }\mathrm{x}\mathrm{ }-\mathrm{ }1\right|\mathrm{ },\mathrm{ }\mathrm{y}=0$ and $\left|\mathrm{ }\mathrm{x}\mathrm{ }\right|=2$ is

Area lying between the curves $\mathrm{y}=\tan \mathrm{x},\mathrm{ }\mathrm{y}=\cot \mathrm{x}$ and $\mathrm{x}$ -axis, $\mathrm{x}\in \left[0,\mathrm{\pi }/2\right]$ is

The area enclosed by the curves $\mathrm{y}=\left|\mathrm{x}\mathrm{ }-1\right|$ and $\mathrm{y}=3-\left|\mathrm{x}\right|$ is-

The area between the parabola $y^2=4ax$ and its latus rectum is: