Function Area Calculation in First Quadrant
We need to find the area enclosed by the function $y = 8 - x$ and the coordinate axes (x-axis and y-axis) specifically within the first quadrant ($x \ge 0, y \ge 0$).
Finding the Boundaries
The boundaries are:
- The line $y = 8 - x$.
- The y-axis ($x=0$).
- The x-axis ($y=0$).
Identifying the Shape and Vertices
To find the area, we first determine where the line intersects the coordinate axes:
- y-intercept: Set $x = 0$ in the equation $y = 8 - x$. This gives $y = 8 - 0 = 8$. The intersection point is $(0, 8)$.
- x-intercept: Set $y = 0$ in the equation $y = 8 - x$. This gives $0 = 8 - x$, so $x = 8$. The intersection point is $(8, 0)$.
The region bounded by the line and the axes in the first quadrant is a right-angled triangle with vertices at the origin $(0, 0)$, the x-intercept $(8, 0)$, and the y-intercept $(0, 8)$.
Area Calculation
The base of the triangle lies along the x-axis and has a length of 8 units. The height of the triangle lies along the y-axis and also has a length of 8 units.
The formula for the area of a triangle is:
$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $
Substitute the base and height values:
$ \text{Area} = \frac{1}{2} \times 8 \times 8 $
$ \text{Area} = \frac{1}{2} \times 64 $
$ \text{Area} = 32 $
The area bounded by the function $y = 8 - x$ and the coordinate axes in the first quadrant is 32 square units.
Important Questions from Application Of Definite Integral (Area)
- Consider the following curve in polar coordinates.
$$ r = 2 - 2 \sin \theta $$
Which one of the following is the area enclosed by the curve for $0 \leq \theta \leq 2\pi$ ? Consider the equation for a curve, $y = f(x) = x^2 + x$.
The area enclosed by the curve, the x -axis ($y = 0$ line); the vertical lines passing through $x = 1$ and $x = 2$ is _________ (rounded off to 2 decimal places)The area bounded by the curves, $y = \sqrt{x}$, and $y = 8x^2$ is _______________(rounded off to 3 decimal places)
- The area of the region (rounded off to one decimal place) enclosed between the curves $y = x$ and $y = 3\sqrt{x}$ and between the lines $x = 0$ and $x = 1$ is ________ units.
- The arc length of the parametric curve: $x = \cos \theta$, $y = \sin \theta$, $z = \theta$ from $\theta = 0$ to $\theta = 2\pi$ is equal to ________ (round off to one decimal place).
