Area Under the Curve — NDA Mathematics PYQs

Directions : For the next two (02) items that follow :
Let $k$ be the area between the curve $y=\sin x$ and x-axis in the interval $\left[0,\frac{\pi }{4}\right]$

What is the area between the curve $y=\sin x$ and the x-axis in the interval $\left[\frac{\pi }{4},\frac{\pi }{2}\right]$ ?

Directions : For the next two (02) items that follow :
Let $k$ be the area between the curve $y=\sin x$ and x-axis in the interval $\left[0,\frac{\pi }{4}\right]$

What is the area between the curve $y=\cos x$ and the x-axis in the interval $\left[\frac{\pi }{4},\frac{\pi }{2}\right]$ ?

Directions : For the next two (02) items that follow :
Consider the curves $y=x^2$ and $y=2|x|$.

What is the number of points of intersection of the curves ?

Directions : For the next two (02) items that follow :
Consider the curves $y=x^2$ and $y=2|x|$.

What is the area bounded by the curves, the lines $x=0$ and $x=1$ ?

Consider the following for the two (02) items that follow :
The slope of the tangent to the curve $y=f(x)$ at $(x,f(x))$ is $4$ for every real number $x$ and the curve passes through the origin.

What is the area bounded by the curve, the $x$-axis and the line $x=4$?

Consider the following for the two (02) items that follow :
Let the curve $f(x)=|x-3|$.

What is the area bounded by the curve $f(x)$ and $y=3$?

Consider the following for the two (02) items that follow :
Let the function $f(x)=x^2-1$.

What is the area bounded by the function $f(x)$ and the $x$-axis?

Directions : For the following three (03) items :
Consider the function $f(x)=x|x|$.

What is the area bounded by the curve $f(x)$, the $x$-axis and the lines $x=-2$ and $x=1$?

What is the area of the region bounded by $|x|\le 2k$ and $|y|\le k$, where $k$ is a positive real number?

*Directions : Consider the following for the next two (02) items that follow :

The circle $x^2+y^2-2x=0$ is partitioned by line $y=x$ in two segments. Let $A_1,A_2$ be the areas of major and minor segments respectively.*

What is the value of $A_1$ ?