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

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In summary, the question is about the value of the integral of sinx from 0 to pi, which is equal to 2. However, the person thought of it in terms of a half circle with radius one, which would make the integral equal to Pi/2. They were mistaken in assuming x as the coordinate instead of the angle. The correct integral would be ∫(sinx)2 dx, which is equal to Pi/2.
  • #1
Karim Habashy
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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 ?
 
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  • #2
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.
 
  • #3
You are right, thanks. the integration would be ∫(sinx)2 dx which is Pi/2.
 

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

1. What is the equation for finding the area of a half circle?

The equation for finding the area of a half circle is A = 1/2 * π * r^2, where A represents the area and r represents the radius of the half circle.

2. How is the integrand for sin(x) derived for finding the area of a half circle?

The integrand for sin(x) is derived by using the formula for the circumference of a circle, C = 2πr, and solving for the length of the arc of the half circle, which is equal to the radius multiplied by the central angle, θ. This gives us the equation x = rθ. By substituting this value into the circumference formula, we get C = 2πx/r. Using the definition of sine, sin(θ) = opposite/hypotenuse, we can then rearrange the equation to get sin(θ) = x/r, which is the integrand for sin(x).

3. How do you solve the integral of sin(x) to find the area of a half circle?

To solve the integral of sin(x), we use the integration by parts method, which involves finding the anti-derivative of sin(x) and multiplying it by the limits of integration. This gives us the equation A = 1/2 * π * r * (-cos(x)) evaluated from 0 to π, which simplifies to A = π * r.

4. Can the same method be used for finding the area of a full circle?

Yes, the same method can be used for finding the area of a full circle. However, the limits of integration would change to 0 and 2π, and the final answer would be 2πr instead of just πr.

5. Are there any other methods for finding the area of a half circle?

Yes, there are other methods for finding the area of a half circle, such as using the formula A = π * (d/2)^2, where d is the diameter of the half circle. This method does not involve integration and is simpler to use for calculating the area of a half circle.

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