The area bounded by the curve y log x, x-axis, ordinates x = 1 and x = 2 in sq. units is
The x-axis is the horizontal number line in a Cartesian coordinate system. It represents the independent variable in graphs and equations. Points on the x-axis have a y-coordinate of zero. It helps in plotting functions and analyzing mathematical relationships in algebra geometry and calculus for various scientific and engineering applications.
Class 12 Maths Chapter 8 Applications of Integrals is an essential topic for the CBSE Exam 2024-25. It focuses on calculating areas between curves and finding volumes of solids using integration. The chapter has practical applications in physics engineering and real-life problems. Mastery of these concepts is crucial for higher studies and understanding complex problems.
The area bounded by the curve y = log x, the x-axis, and the ordinates x = 1 and x = 2 can be found using the definite integral of log x from x = 1 to x = 2.
Step 1: Write the integral
The area is calculated as:
A = ∫₁² log x dx
Step 2: Integrate
This can be evaluated as a result of integration by parts. Note that the integration by parts formula is as follows:
∫ u dv = uv – ∫ v du
Here let u = log x and dv = dx, then du = (1/x) dx, v = x
Thus
∫ log x dx = x log x – ∫ x (1/x) dx = x log x – x
Step 3: Calculate the result
Now integrate the integral from x = 1 to x = 2:
A = [x log x – x]₁²
When x = 2,
2 log 2 – 2
When x = 1,
1 log 1 – 1 = 0 – 1 = -1
Hence area is:
A = (2 log 2 – 2) – (-1) = 2 log 2 – 2 + 1 = 2 log 2 – 1
Now substituting the value for log 2 = 0.693.
A ≈ 2(0.693) – 1 = 1.386 – 1 = 0.386
Click here for more:
https://www.tiwariacademy.com/ncert-solutions/class-12/maths/#chapter-8