Understanding the Integral of the Semicircle and Its Relationship to Pi

The value of π (pi) is one of the most well-known mathematical constants, and its relationship to circles and semicircles is well established.

Integrating the Semicircle

Consider the integral (int_{-1}^{1} sqrt{1 - x^2} , dx). This integral represents the area under the curve of a semicircle with radius 1. To understand this, we first need to recall the equation of a circle and then analyze the semicircle specifically.

The Equation of a Circle

The equation of a circle with radius 1 is given by:

(x^2 y^2 1)

When considering the top (or bottom) half of the circle, we solve for (y):

(y sqrt{1 - x^2})

Graphical Interpretation of the Integral

The integral given above calculates the area under the curve (y sqrt{1 - x^2}) from (-1) to (1). This area corresponds to the positive part of the semicircle, which is half of the full circle.

Area Calculation

The area of a full circle with radius 1 is given by the formula:

(pi r^2 pi (1)^2 pi)

Since the integral represents half of the circle, the result of the integral from (-1) to (1) is:

(frac{pi}{2})

To get the area of the full semicircle, we need to multiply this result by 2:

(2 times frac{pi}{2} pi)

Conclusion

The integral (int_{-1}^{1} sqrt{1 - x^2} , dx) effectively calculates the area of a semicircle with radius 1, which is equal to (pi). This relationship between the integral and the value of (pi) demonstrates a fundamental connection in calculus and geometry.

Related Keywords: integral, semicircle, pi