Compute this definite integral using geometric methods: $\int _ { - 5 } ^ { 5 } - \sqrt { 25 - x ^ { 2 } } d x$ .

A) $25 \pi / 2$
B) $25 \pi$
C) $- 25 \pi / 2$
D) $- 25 \pi$
E) $50 \pi$

Question

Compute this definite integral using geometric methods: $\int _ { - 5 } ^ { 5 } - \sqrt { 25 - x ^ { 2 } } d x$ .

A)  $25 \pi / 2$
B) $25 \pi$
C) $- 25 \pi / 2$
D) $- 25 \pi$
E) $50 \pi$

Quizplus · Accepted Answer

The integral $\int _ { - 5 } ^ { 5 } - \sqrt { 25 - x ^ { 2 } } d x$ represents the negative area of a semicircle with radius 5 centered at the origin. The area of a full circle with radius 5 is $25\pi$, so half of that area is $25\pi / 2$, and since the integral is of the negative semicircle, the answer is $- 25 \pi / 2$.

Compute This Definite Integral Using Geometric Methods ∫−55−25−x2dx\int _ { - 5 } ^ { 5 } - \sqrt { 25 - x ^ { 2 } } d x∫−55​−25−x2​dx

Compute This Definite Integral Using Geometric Methods $\int _ { - 5 } ^ { 5 } - \sqrt { 25 - x ^ { 2 } } d x$