Applications of integrals involve calculating areas under curves, volumes of solids and solving problems related to physics and engineering. They help in determining quantities such as distance, work and pressure. Integrals are used in various fields to model real-world phenomena, including motion, fluid dynamics and electrical circuits.
Class 12 Maths Chapter 8 Applications of Integrals is an important topic for the CBSE Exam 2024-25. It focuses on calculating areas between curves and volumes of solids using integration. The chapter also covers real-life applications in fields like physics and engineering. Understanding these concepts is crucial for solving complex problems and higher studies.
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We have to calculate the definite integral of sin x from 0 to 2π in order to find the area bounded by the curve y = sin x, the x-axis, and the ordinates x = 0 and x = 2π.
Step 1: Setup the integral
The area is given by:
A = ∫₀²π sin x dx
Step 2: Solve the integral
We know the integral of sin x is
∫ sin x dx = -cos x
Evaluate the integral from 0 to 2π:
A = [-cos x]₀²π
At x = 2π:
-cos(2π) = -1
At x = 0:
-cos(0) = -1
Hence, area is:
A = -1 – (-1) = 0
Since sin x is above the x-axis for the interval [0, π] and below the x-axis for the interval [π, 2π], the areas of these two parts are equal in magnitude but opposite in sign. So we take the absolute value of the integrals over both intervals.
Step 3: Evaluate the area with absolute value
Area =
A = 2 × ∫₀π sin x dx = 2 [-cos x]₀π
At x = π:
-cos(π) = 1
At x = 0:
-cos(0) = -1
Thus, the area is: A = 2 × (1 – (-1)) = 2 × 2 = 4
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