Math (Definite Integrals)

Sketch the region given by the definite integral. Use geometric shapes and formulas to evaluate the integral (a > 0, r > 0).

r
∫ sqrt(r^2 - x^2) dx
-r

While I recognize that this looks similar to a circle function, I'm not sure how to graph and evaluate this definite integral because the lower/upper bounds is throwing me off because it's not giving us a number, but a variable.

How could we approach this problem? I know there's tons of calculators that can calculate the integral by finding the anti derivative, but this is not the case for this problem specifically.

Any help is greatly appreciated!

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  1. Should we assume r = 1?

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  2. Do not assume r=1. But, it is a constant. And you are right, the graph is a sem-circle, so the area is π/2 r^2

    Not sure where the a comes in, but to do the integral, now is the time to start getting used to trig substitutions. Let

    x = r sinθ
    r^2-x^2 = r^2(1-sin^2θ) = r^2 cos^2θ
    dx = r cosθ dθ

    Now you have

    ∫[-π/2,π/2] r cosθ (r cosθ dθ)
    = r^2 ∫[-π/2,π/2] cos^2θ dθ
    = r^2/2 ∫[-π/2,π/2] (1+cos2θ) dθ
    = r^2/2 (θ + 1/2 sin2θ) [[-π/2,π/2]
    = r^2/2 [(π/2+0)-(-π/2+0)]
    = πr^2/2

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