Question:
An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :
An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :
Show Hint
Geometrically, this region is a trapezoid with vertices \( (0,0), (1,0), (1,-2), (0,-4) \). You can verify the area using the formula: \( \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} = \frac{1}{2} \times (4 + 2) \times 1 = 3 \).
Updated On: Sep 10, 2026
- \( 1 \text{ sq. unit} \)
- \( 3 \text{ sq. units} \)
- \( 2 \text{ sq. units} \)
- \( 4 \text{ sq. units} \)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
• Area under a curve \( y = f(x) \) bounded by the x-axis and lines \( x = a \), \( x = b \) is given by \( \int_a^b |f(x)| \, dx \).
• For a region below the x-axis, the definite integral will be negative, so we take the absolute value for area.
Step 1: Determine the integration limits and sign of the function
The area is bounded by the y-axis (\( x = 0 \)), the line \( x = 1 \), and the x-axis (\( y = 0 \)).
In the interval \( [0, 1] \), let's check the sign of \( y = 2x - 4 \):
At \( x = 0, y = -4 \).
At \( x = 1, y = -2 \).
Since the function is always negative in this interval, the curve lies below the x-axis.
Step 2: Set up the integral for the area
Area \( A = \int_0^1 |2x - 4| \, dx \).
Since \( 2x - 4 \) is negative on \( [0, 1] \), \( |2x - 4| = -(2x - 4) = 4 - 2x \).
\[ A = \int_0^1 (4 - 2x) \, dx \]
Step 3: Evaluate the definite integral
Integrate the expression:
\[ A = [4x - x^2]_0^1 \]
Substitute the limits:
\[ A = (4(1) - (1)^2) - (4(0) - 0^2) \]
\[ A = (4 - 1) - 0 = 3 \text{ sq. units} \]
• Area under a curve \( y = f(x) \) bounded by the x-axis and lines \( x = a \), \( x = b \) is given by \( \int_a^b |f(x)| \, dx \).
• For a region below the x-axis, the definite integral will be negative, so we take the absolute value for area.
Step 1: Determine the integration limits and sign of the function
The area is bounded by the y-axis (\( x = 0 \)), the line \( x = 1 \), and the x-axis (\( y = 0 \)).
In the interval \( [0, 1] \), let's check the sign of \( y = 2x - 4 \):
At \( x = 0, y = -4 \).
At \( x = 1, y = -2 \).
Since the function is always negative in this interval, the curve lies below the x-axis.
Step 2: Set up the integral for the area
Area \( A = \int_0^1 |2x - 4| \, dx \).
Since \( 2x - 4 \) is negative on \( [0, 1] \), \( |2x - 4| = -(2x - 4) = 4 - 2x \).
\[ A = \int_0^1 (4 - 2x) \, dx \]
Step 3: Evaluate the definite integral
Integrate the expression:
\[ A = [4x - x^2]_0^1 \]
Substitute the limits:
\[ A = (4(1) - (1)^2) - (4(0) - 0^2) \]
\[ A = (4 - 1) - 0 = 3 \text{ sq. units} \]
Was this answer helpful?
0
0
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
View More Questions
Top CBSE CLASS XII Area under Simple Curves Questions
The area of the region enclosed between the curve \( y = |x| \), x-axis, \( x = -2 \)} and \( x = 2 \) is:
- The area of the region bounded by the curve \(y = x\) and the x-axis, between the boundaries \(x = 0\) and \(x = 2\) is:
- The area of the region bounded by the curve \(y = x\) and x-axis, between \(x = 0\) and \(x = 2\) is :
- An ant is observed crawling on a sheet of paper along a straight line given by equation \( y = 2x - 4 \). Area of the surface covered by the ant bounded by y-axis, x-axis and \( x = 1 \) is :
- Sketch the curve \( \{(x, y) : 100x^2 + 25y^2 = 2500\ \) and find the area of the region enclosed by it, using integration.}
View More Questions
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
View More Questions