Greatest Integer Function Area Calculation on $[1,4]$
The function is defined as $y = [x]$, where $[x]$ represents the greatest integer less than or equal to $x$. The task is to find the area under this curve for the interval $x \in [1, 4]$.
Area Breakdown by Interval
The interval $[1, 4]$ can be divided into subintervals where the value of $[x]$ remains constant:
- Interval $[1, 2)$: Here, $y = [x] = 1$. The area contribution is calculated as Base $\times$ Height = $(2-1) \times 1 = 1 \times 1 = 1$.
- Interval $[2, 3)$: Here, $y = [x] = 2$. The area contribution is Base $\times$ Height = $(3-2) \times 2 = 1 \times 2 = 2$.
- Interval $[3, 4)$: Here, $y = [x] = 3$. The area contribution is Base $\times$ Height = $(4-3) \times 3 = 1 \times 3 = 3$.
Total Area Computation
The total area under the curve $y = [x]$ from $x=1$ to $x=4$ is the sum of the areas calculated for each subinterval. This can be represented as the definite integral:
$ \text{Total Area} = \int_{1}^{4} [x] \, dx = \int_{1}^{2} 1 \, dx + \int_{2}^{3} 2 \, dx + \int_{3}^{4} 3 \, dx $
Summing the individual areas:
$ \text{Total Area} = 1 + 2 + 3 = 6 $
Therefore, the area under $y = [x]$ for $x \in [1, 4]$ is 6 square units.
Important Questions from Area Under Curve
- A function $y(x)$ is defined in the interval $[0, 1]$ on the x-axis as
$y(x) = \begin{cases} 2 & \text{if } 0 \le x < \frac{1}{3} \\ 3 & \text{if } \frac{1}{3} \le x < \frac{3}{4} \\ 1 & \text{if } \frac{3}{4} \le x \le 1 \end{cases}$
Which one of the following is the area under the curve for the interval $[0, 1]$ on the x-axis? - The area of the region bounded by the parabola $x = -y^2$ and the line $y = x + 2$ equals
The work done by the force $F = (x + y)\hat{i} - (x^2 + y^2)\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in $\vec{OX}$ and $\vec{OY}$ directions, respectively, along the upper half of the circle $x^2 + y^2 = 1$ from $(1,0)$ to $(-1,0)$ in the $xy$-plane is
In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.
The probability that any point picked randomly within the square falls in the shaded area is ___________.- If $f(x) = 2 \ln(\sqrt{e^x})$, what is the area bounded by $f(x)$ for the interval $[0, 2]$on the x-axis?
