Trapezoidal Rule Area Calculation
To find the area under the curve $y = x^2 + 2x$ between $x = 0$ and $x = 4$ using the trapezoidal rule with a step size $h = 1$, we follow these steps:
1. Define Points and Calculate Function Values
The interval is $[0, 4]$ and the step size $h=1$. The points are $x_0 = 0, x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4$. We calculate the corresponding $y$ values using $y = x^2 + 2x$:
- $y_0 = f(0) = 0^2 + 2(0) = 0$
- $y_1 = f(1) = 1^2 + 2(1) = 1 + 2 = 3$
- $y_2 = f(2) = 2^2 + 2(2) = 4 + 4 = 8$
- $y_3 = f(3) = 3^2 + 2(3) = 9 + 6 = 15$
- $y_4 = f(4) = 4^2 + 2(4) = 16 + 8 = 24$
2. Apply Trapezoidal Rule Formula
The trapezoidal rule formula for approximating the area is:
Area $\approx \frac{h}{2} [y_0 + 2(y_1 + y_2 + ... + y_{n-1}) + y_n]$
Here, $h=1$ and $n=4$. Plugging in the values:
Area $\approx \frac{1}{2} [y_0 + 2(y_1 + y_2 + y_3) + y_4]$
Area $\approx \frac{1}{2} [0 + 2(3 + 8 + 15) + 24]$
3. Calculate the Area
Perform the calculation:
Area $\approx \frac{1}{2} [0 + 2(26) + 24]$
Area $\approx \frac{1}{2} [52 + 24]$
Area $\approx \frac{1}{2} [76]$
Area $\approx 38$
The calculated area using the trapezoidal rule is 38.
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?
