Question

use geometry (not riemann sums) to evaluate the definite integral. sketch the graph of the integrand, show the region in question, and interpret your result. $int_{-5}^{0}sqrt{25 - x^{2}}dx$. choose the correct graph below.

Explanation:

Step1: Identify the curve

The integrand \( y = \sqrt{25 - x^2} \) squares to \( x^2 + y^2 = 25 \), a circle with radius 5, center at origin; \( y \geq 0 \) gives upper semicircle.

Step2: Determine the region

Integration limits \( x \in [-5, 0] \): left half of upper semicircle, a quarter-circle (1/4 of full circle).

Step3: Calculate area

Area of full circle: \( \pi r^2 = 25\pi \). Quarter-circle area: \( \frac{25\pi}{4} \).

Answer:

The value of the integral is \( \frac{25\pi}{4} \), corresponding to the area of a quarter-circle. The correct graph is A.