Integrating Sin(x): Solve the Area of a Half Circle

  • Context: Undergrad 
  • Thread starter Thread starter Karim Habashy
  • Start date Start date
  • Tags Tags
    Integration
Click For Summary
SUMMARY

The discussion centers on the integration of the sine function, specifically the definite integral ∫sin(x) dx from 0 to π, which equals 2. The confusion arises from interpreting this integral as the area of a half circle with a radius of one, which is actually π/2. The correct approach involves recognizing that the integration represents the area under the sine curve, not the area of a circle. The final conclusion is that the correct integral for the area of the half circle is ∫(sin(x))^2 dx, yielding π/2.

PREREQUISITES
  • Understanding of definite integrals in calculus
  • Familiarity with the sine function and its properties
  • Knowledge of the geometric interpretation of integrals
  • Basic concepts of area calculation in geometry
NEXT STEPS
  • Study the properties of definite integrals in calculus
  • Learn about the geometric interpretation of trigonometric integrals
  • Explore the derivation of the area of a circle and its relationship to integrals
  • Investigate the use of integration techniques for trigonometric functions
USEFUL FOR

Students of calculus, mathematics educators, and anyone interested in understanding the relationship between trigonometric functions and geometric areas.

Karim Habashy
Messages
31
Reaction score
1
Hi All,

This is a very elementary question but, from calculus :

∫sinx dx from 0 to pi = 2, but i thought of it, in terms of the attached driagram.
upload_2016-3-5_19-42-31.png


And if we think of it, this way, the integration is the area of a half circle of radius one and is equal to Pi/2.

Where did i go wrong ?
 
Physics news on Phys.org
Karim Habashy said:
And if we think of it, this way, the integration is the area of a half circle of radius one and is equal to Pi/2.
No, it's not. Your mistake was to assume ##x## as the coordinate in the horizontal line, when it's actually the angle subtended between the line and the horizontal axis.
 
You are right, thanks. the integration would be ∫(sinx)2 dx which is Pi/2.
 

Similar threads

High School Question about change of variables
  • · Replies 29 ·
Replies
29
Views
5K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Why is the integral of ##\arcsin(\sin(x))## so divisive?
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Calculating the surface area of a sphere using dA
  • · Replies 33 ·
2
Replies
33
Views
7K
Undergrad What's my mistake in this integration problem?
  • · Replies 8 ·
Replies
8
Views
3K
High School Question about volume of a sphere
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Having trouble understanding a seemingly simple u-substitution
  • · Replies 5 ·
Replies
5
Views
2K
High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad Negative area above x-axis from integrating x^2?
  • · Replies 4 ·
Replies
4
Views
3K