Graph the integrand and use geometry to evaluate the integral.

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BlackTom AI · Accepted Answer

The problem asks you to graph the integrand and use geometry to evaluate the integral ∫_{-5}^{5} sqrt(25 - x^2) dx.
First, recognize that y = sqrt(25 - x^2) describes the upper semicircle of radius 5, since x^2 + y^2 = 25 with y ≥ 0.
The definite integral from -5 to 5 corresponds to the area under this semicircular curve above the x-axis over the interval [-5, 5].
Geometrically, this area is exactly half the area of a full circle with radius 5. The area of a full circle is πr^2 = π(5^2) = 25π, so half of that is 25π/2.
Therefore, the value of the integral is 25π/2.
Note on the provided options: if the available choices include numbers without π (e.g., 25/2) or solely π, those would be incorrect because the correct result must include the factor π from the circular area calculation.
In this particular item, the options list is not shown, so I cannot identify which choice corresponds to 25π/2. Nevertheless, the correct evaluation from the geometry of a semicircle is 25π/2.

Graph the integrand and use geometry to evaluate the integral.

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