The area of curve xy = 4, bounded by the lines x =1 and x = 3 and x-axis in sq. units is
The area under a curve represents the total accumulated quantity between the curve and the x-axis over a specific interval. It is calculated using definite integrals. This concept is used to find areas in geometry and applications in physics such as distance, work and probability.
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 techniques. This chapter has practical applications in fields such as physics and engineering. Understanding these concepts is vital for solving real-life problems and further studies.
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Determine the area of the region bounded by the curve xy = 4 and the lines x = 1, x = 3, and the x-axis. First we rewrite the equation xy = 4 to solve for y in terms of x.
Now solving the above equation for y we get,
y = 4/x
Step 1
We begin by setting up the integral
The area can be calculated by integrating y = 4/x from x = 1 to x = 3:
A = ∫₁³ (4/x) dx
Step 2: Evaluate the integral
We know that:
∫ (1/x) dx = ln |x|
Thus, the integral becomes:
A = 4 ∫₁³ (1/x) dx = 4 [ln x]₁³
Step 3: Calculate the area
At x = 3:
ln 3
At x = 1:
ln 1 = 0
Therefore, the area is
A = 4 (ln 3 – 0) = 4 ln 3
Using logarithmic properties:
A = ln (3⁴) = ln 81
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