Graphical Significance of Integral of Sin x
The integral of sin(x) from ( a ) to ( b ) has graphical significance in terms of calculating the area under the curve within this interval. Let’s explore the graphical significance using both the definite integral method and the geometrical method.
Definite Integral Method
The integral of sin(x) from ( a ) to ( b ) is given by:
[Tex]\int_{a}^{b} \sin(x) \,dx = -\cos(x) \Big|_{a}^{b} [/Tex] = -cos(b) + cos(a)
This represents the signed area between the curve sin(x) and the x-axis from ( a ) to ( b ).
Geometrical Method
Consider the graph of sin(x) from ( a ) to ( b ). The area under the curve can be divided into two regions:
- Positive Area: Regions where sin(x) is positive (above the x-axis). This contributes to the positive area under the curve.
- Negative Area: Regions where sin(x) is negative (below the x-axis). This contributes to the negative area under the curve.
The total area is the algebraic sum of these positive and negative areas.
Example:
To find the area under the curve of sin(x) from ( a = 0 ) to ( b = π/2 ).
Using the definite integral method:
∫0π/2 sin x = [-cos x]0π/2 = -cos(π/2) – (-cos 0) = 0 + 1 = 1
This is the signed area under the curve.
Using the geometrical method:
The graph of sin(x) from 0 to (π/2) is a quarter of a circle, and the area is indeed 1.
Integral of Sin x
Integral of sin x is -cos(x) plus a constant (C). It represents the area under the sine curve. The function repeats every 2π radians due to its periodic nature. This article explains the integral of the sine function, showing its formula, proof, and application in finding specific definite integrals. Further, it mentions solved problems and frequently asked questions.
Table of Content
- What is Integral of Sin x?
- Integral of Sin x Formula
- Graphical Significance of Integral of Sin x
- Integral of Sin x Proof by Substitution Method
- Definite Integral of Sin x
- Integral of Sin x From 0 to π
- Integral of Sin x From 0 to π/2
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