Calculate Integral of Sqrt 8-X 2

The integral of √(8 - x²) is a common calculus problem that appears in physics, engineering, and geometry. This page provides a step-by-step guide to calculating it, along with practical applications and common questions.

How to Calculate the Integral of √(8 - x²)

The integral of √(8 - x²) can be evaluated using trigonometric substitution. Here's a step-by-step approach:

Identify the form of the integrand: √(a² - x²) where a² = 8.
Make the substitution x = a sinθ = 2√2 sinθ.
Calculate dx = 2√2 cosθ dθ.
Substitute into the integral: ∫√(8 - (2√2 sinθ)²) * 2√2 cosθ dθ.
Simplify the expression inside the square root: √(8 - 8 sin²θ) = √[8(1 - sin²θ)] = 2√2 √(cos²θ).
Simplify the integral: 2√2 ∫2√2 cosθ * 2√2 cosθ dθ = 8 ∫cos²θ dθ.
Use the identity cos²θ = (1 + cos2θ)/2 to rewrite the integral.
Integrate term by term: 8 ∫(1 + cos2θ)/2 dθ = 4 ∫(1 + cos2θ) dθ = 4(θ + (sin2θ)/2) + C.
Back-substitute θ = arcsin(x/(2√2)) and simplify.

Note: The exact form of the antiderivative is complex, but it can be expressed in terms of inverse trigonometric functions.

Formula Used

The integral of √(8 - x²) from a to b is given by:

∫√(8 - x²) dx = x√(8 - x²)/2 + 4 arcsin(x/(2√2)) + C

This formula is derived using trigonometric substitution and simplifies to a combination of algebraic and inverse trigonometric terms.

Worked Example

Example Calculation

Calculate ∫√(8 - x²) dx from x = 0 to x = 2√2.

Apply the antiderivative formula at the bounds.
At x = 2√2: (2√2)(0)/2 + 4 arcsin(1) = 0 + 4(π/2) = 2π.
At x = 0: 0 + 4 arcsin(0) = 0.
Subtract the lower bound from the upper bound: 2π - 0 = 2π.

The definite integral evaluates to 2π.

Practical Applications

The integral of √(8 - x²) appears in various fields:

Physics: Calculating areas of circular segments.
Engineering: Determining the length of a curve.
Geometry: Finding the area under a semicircle.

Application	Description
Area Calculation	Determines the area under the curve √(8 - x²) between two points.
Curve Length	Used to find the length of the curve y = √(8 - x²).

FAQ

What is the integral of √(8 - x²)?: The integral of √(8 - x²) is x√(8 - x²)/2 + 4 arcsin(x/(2√2)) + C.
How do I evaluate the definite integral of √(8 - x²)?: Use the antiderivative formula and evaluate at the upper and lower bounds, then subtract.
What is the area under √(8 - x²) from 0 to 2√2?: The area is 2π, representing the area of a semicircle with radius 2√2.
Can I use this formula for other similar integrals?: Yes, the method applies to integrals of the form √(a² - x²) with appropriate substitutions.