A curve is a continuous and smooth line that can bend or change direction without sharp angles. In mathematics, curves represent the graphical representation of functions or relations. They are used to model various phenomena in geometry, physics and engineering, describing movement, growth and other real-world situations.
Class 12 Maths Chapter 8 Applications of Integrals is an important topic for the CBSE Exam 2024-25. It deals with calculating areas between curves and volumes of solids using integration. The chapter has practical applications in physics and engineering. Understanding these concepts is crucial for solving real-life problems and higher studies.
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To find the area bounded by the curve y = sin x, the x-axis, and the ordinates x = 0 and x = 2, we must calculate the definite integral of sin x from 0 to 2.
Step 1: Set up the integral
The area is given by:
A = ∫₀² sin x dx
Step 2: Solve the integral
We know the integral of sin x is:
∫ sin x dx = -cos x
Now, evaluate the integral from 0 to 2:
A = [-cos x]₀²
At x = 2:
-cos(2)
At x = 0:
-cos(0) = -1
Thus, the area is:
A = -cos(2) – (-1) = 1 + cos(2)
Step 3: Approximate the result
Using a calculator, cos(2) ≈ -0.416, so:
A ≈ 1 – (-0.416) = 1 + 0.416 = 1.416
Thus, the closest option to this result is: 4/3 sq. units.
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