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Math  /  Calculus

QuestionEvaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using geometry. ∫−5552−x2dx\int_{-5}^{5} \sqrt{5^{2}-x^{2}} d x □\square

Studdy Solution

STEP 1

1. The integral represents the area under the curve y=52−x2 y = \sqrt{5^2 - x^2} from x=−5 x = -5 to x=5 x = 5 .
2. The function y=52−x2 y = \sqrt{5^2 - x^2} describes a semicircle with radius 5 centered at the origin.

STEP 2

1. Identify the geometric shape represented by the integral.
2. Calculate the area of the geometric shape.

STEP 3

Recognize that the function y=52−x2 y = \sqrt{5^2 - x^2} is the equation of the upper half of a circle centered at the origin with radius 5. This is because the equation x2+y2=52 x^2 + y^2 = 5^2 describes a circle with radius 5, and y=52−x2 y = \sqrt{5^2 - x^2} is the positive square root, representing the upper semicircle.

STEP 4

The integral ∫−5552−x2 dx \int_{-5}^{5} \sqrt{5^2 - x^2} \, dx represents the area under the upper semicircle from x=−5 x = -5 to x=5 x = 5 . The area of a full circle with radius 5 is given by the formula πr2 \pi r^2 . Therefore, the area of the semicircle is half of that:
Area of full circle=π×52=25π \text{Area of full circle} = \pi \times 5^2 = 25\pi Area of semicircle=12×25π=25π2 \text{Area of semicircle} = \frac{1}{2} \times 25\pi = \frac{25\pi}{2}
The area represented by the integral is:
25π2 \boxed{\frac{25\pi}{2}}

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