Two straight lines pass through the origin $(x_0, y_0) = (0,0)$. One of them passes through the point $(x_1, y_1) = (1,3)$ and the other passes through the point $(x_2, y_2) = (1,2)$. What is the area enclosed between the straight lines in the interval $[0, 1]$ on the x-axis?
Finding Line Equations
Both lines pass through the origin $(0,0)$. The equation of a line passing through the origin is given by $y = mx$, where $m$ is the slope.
- Line 1 passes through $(0,0)$ and $(1,3)$. The slope $m_1$ is calculated as: $m_1 = \frac{y_1 - y_0}{x_1 - x_0} = \frac{3 - 0}{1 - 0} = 3$. The equation for Line 1 is $y = 3x$.
- Line 2 passes through $(0,0)$ and $(1,2)$. The slope $m_2$ is calculated as: $m_2 = \frac{y_2 - y_0}{x_2 - x_0} = \frac{2 - 0}{1 - 0} = 2$. The equation for Line 2 is $y = 2x$.
Calculating Enclosed Area
The area enclosed between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x=b$, where $f(x) \geq g(x)$ in the interval, is given by the definite integral: $ A = \int_{a}^{b} (f(x) - g(x)) dx $ In the interval $[0, 1]$ on the x-axis, $3x \geq 2x$. Therefore, the upper curve is $y = 3x$ and the lower curve is $y = 2x$. The interval is $[0, 1]$.
The area $A$ is:
$ A = \int_{0}^{1} (3x - 2x) dx $ $ A = \int_{0}^{1} x dx $Evaluating the Integral
Now, we evaluate the definite integral:
$ A = \left[ \frac{x^2}{2} \right]_{0}^{1} $ $ A = \frac{(1)^2}{2} - \frac{(0)^2}{2} $ $ A = \frac{1}{2} - 0 $ $ A = 0.5 $The area enclosed between the straight lines in the interval $[0, 1]$ on the x-axis is $0.5$ square units.
Important Questions from Application Of Definite Integral (Area)
The equation of a closed curve in two-dimensional polar coordinates is given by $r = \frac{2}{\sqrt{\pi}}(1 - \sin \theta)$. The area enclosed by the curve is ______ (answer in integer).
The area bounded by the curves, $y = \sqrt{x}$, and $y = 8x^2$ is _______________(rounded off to 3 decimal places)
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 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.
- Let $S_1$ be the plane figure consisting of the points $(x, y)$ given by the inequalities $|x-1| \le 2$ and$|y + 2| \le 3$. Let $S_2$ be the plane figure given by the inequalities $x - y \ge -2$, $y \ge 1$, and $x \le 3$.Let $S$ be the union of $S_1$ and $S_2$. The area of $S$ is
