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FUNKER
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how would you integrate
(r^2-x^2)^(1/2)
where r is constant
thanks peace
(r^2-x^2)^(1/2)
where r is constant
thanks peace
The formula for integrating (r^2-x^2)^(1/2) is ∫(r^2-x^2)^(1/2)dx = (x/2)*(√(r^2-x^2) + r^2*sin^(-1)(x/r)) + C, where C is the constant of integration.
The domain of integration for (r^2-x^2)^(1/2) is [-r, r] as the square root term becomes imaginary for values of x outside of this range.
Yes, (r^2-x^2)^(1/2) can be solved using the substitution u = √(r^2-x^2), which transforms the integral into ∫u^2*(1-u^2/r^2) dx. This can then be solved by using the trigonometric substitution u = r*sin(θ) and applying the Pythagorean identity.
The integration of (r^2-x^2)^(1/2) can represent the area under a semicircle with radius r, where r is the distance from the center of the circle to the x-axis. This can be useful in calculating the area of a curved surface or in problems involving circular motion.
Yes, the integration of (r^2-x^2)^(1/2) can also be solved using integration by parts, by setting u = (r^2-x^2)^(1/2) and dv = dx. This method may be useful in certain cases where substitution is not applicable or more complicated.