Match the following : (A) Area enclosed by y=|x|,|x|=1 and...

Match the following : (A) Area enclosed by $y=|x|,|x|=1$ and (p) $\quad 2$ $y=0$ is (B) Area enclosed by the curve $y=\sin x$, (q) 4 $x=0, x=\pi$ and $y=0$ is (C) If the area of the region bounded by (r) 27 $x^{2} \leq y$ and $y \leq x+2$ is $\frac{k}{4}$, then $k$ is equal to (D) Area of the quadrilateral formed by (s) 18 tangents at the ends of latusrectum of ellipse of ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$ is

Match the following :
(A) Area enclosed by $y=|x|,|x|=1$ and
(p) $\quad 2$ $y=0$ is
(B) Area enclosed by the curve $y=\sin x$,
(q) 4 $x=0, x=\pi$ and $y=0$ is
(C) If the area of the region bounded by
(r) 27
$x^{2} \leq y$ and $y \leq x+2$ is $\frac{k}{4}$, then $k$ is equal to
(D) Area of the quadrilateral formed by
(s) 18
tangents at the ends of latusrectum of ellipse of ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$ is

Key Concepts

Tangents to Curves

Tangents to curves are lines that touch the curve at exactly one point and reflect the instantaneous direction of the curve. They are often used to form additional geometric figures, such as quadrilaterals, and understanding their derivation and properties is key for calculating areas determined by these tangent segments.

Ellipse Geometry

Ellipse geometry pertains to the study of ellipses defined by standard equations, revealing important properties such as axes, focal points, and the latus rectum. These characteristics are crucial for solving problems involving tangents, chords, and areas related to ellipses.

Inequalities and Bounded Regions

Using inequalities to define regions involves identifying the area where one function dominates another or where several functions intersect. This concept is essential in setting up correct integration limits and ensuring that the computed area corresponds exactly to the geometrically defined region.

Graphical Analysis

Graphical analysis involves examining the visual representations of functions to understand their behavior, symmetry, and points of intersection. This skill is crucial for correctly identifying the boundaries of regions whose areas are to be computed and for setting up the appropriate integrals.

Properties of the Absolute Value Function

The absolute value function produces a V-shaped graph that is symmetric with respect to the y-axis. Knowledge of its piecewise linear nature helps in dividing the domain appropriately to calculate areas, particularly when the function is confined or compared to a constant value.

Definite Integration

Definite integration is the process of computing the area under a curve or between curves using integrals with specified limits. It is fundamental in calculating exact areas enclosed by various functions, such as absolute value functions, trigonometric functions, and others, by summing infinitesimal contributions over an interval.

Trigonometric Function Characteristics

Trigonometric functions, such as the sine function, are periodic and have well-defined behaviors over specific intervals. Understanding these properties, including their symmetry and zeros, is essential for accurately determining the areas beneath or between these curves over a given period.

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