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. $$\int_{-5}^{5} \sqrt{5^{2}-x^{2}} d x$$ $\square$

Studdy Solution
The integral $\int_{-5}^{5} \sqrt{5^2 - x^2} \, dx$ represents the area under the upper semicircle from $x = -5$ to $x = 5$. The area of a full circle with radius 5 is given by the formula $\pi r^2$. Therefore, the area of the semicircle is half of that:
$$\text{Area of full circle} = \pi \times 5^2 = 25\pi$$ $$\text{Area of semicircle} = \frac{1}{2} \times 25\pi = \frac{25\pi}{2}$$
The area represented by the integral is:
$$\boxed{\frac{25\pi}{2}}$$