A curve is a continuous and flowing line without sharp angles that extends in a plane or space. It can be open or closed and is defined mathematically by equations. Curves represent graphs of functions and are used in geometry physics and engineering to describe motion shapes and various natural phenomena.
Class 12 Maths Chapter 8 Applications of Integrals is an important topic for the CBSE Exam 2024-25. It focuses on finding areas bounded by curves and calculating volumes of solids. It is useful in solving real-life problems related to physics and engineering. Understanding these concepts is essential for higher studies and practical applications.
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To determine the area bounded by the curve y = 2ˣ, the x-axis, and the ordinates x = 0 and x = 4, we are required to compute the definite integral of 2ˣ from x = 0 to x = 4.
Step 1: Write down the integral
The area is given by:
A = ∫₀⁴ 2ˣ dx
Step 2: Evaluate the integral
The integral of 2ˣ is:
∫ 2ˣ dx = (2ˣ) / ln 2
Now, calculate the area under the curve from x = 0 to x = 4:
A = [(2ˣ) / ln 2]₀⁴
At x = 4:
(2⁴) / ln 2 = 16 / ln 2
At x = 0:
(2⁰) / ln 2 = 1 / ln 2
Therefore, the area:
A = (16 / ln 2) – (1 / ln 2) = 15 / ln 2
Step 3: Final result
So the region is:
A = 15 / ln 2 square units
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