The area of the region bounded by the curve y = √x -1 and the lines x = 1 and x = 5 is
A curve is a continuous and smooth flowing line that extends in a plane or space without sharp angles. It is represented mathematically by equations and can be open or closed. Curves are used in geometry physics and engineering to describe motion shapes and various natural or man-made structures.
Class 12 Maths Chapter 8 Applications of Integrals is an important topic for the CBSE Exam 2024-25. It focuses on finding areas enclosed by curves and calculating volumes of solids using integration. These concepts have practical applications in physics and engineering. Understanding them is essential for solving real-life problems and higher studies.
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To determine the area of the region bounded by the curve y = √x – 1 and the lines x = 1 and x = 5, we must set up the definite integral.
The area is given by:
A = ∫₁⁵ (√x – 1) dx
Step 1: Solve the integral
We can split the integral into two parts:
A = ∫₁⁵ √x dx – ∫₁⁵ 1 dx
First integral:
∫ √x dx = ∫ x^(1/2) dx = (2/3) x^(3/2)
Evaluating this from 1 to 5:
[(2/3) x^(3/2)]₁⁵ = (2/3) (5^(3/2) – 1^(3/2)) = (2/3) (5√5 – 1)
Second integral:
∫ 1 dx = x
Evaluating this from 1 to 5:
[x]₁⁵ = 5 – 1 = 4
Step 2: Combine the results
The total area is:
A = (2/3) (5√5 – 1) – 4
Simplifying this expression gives the final result:
A = 13/3 square units
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