Solve the problem.
-Find the volume generated by revolving about the $x$-axis the region bounded by the following graph: $y = \sqrt { 49 - x ^ { 2 } } , x = 0 , \quad x = 7$
A) $14 \pi$
B) $\frac { 686 } { 3 } \pi$
C) $\frac { 1372 } { 3 } \pi$
D) $196 \pi$

Question

Quizplus · Accepted Answer

The volume generated by revolving the given region around the x-axis can be found using the formula for the volume of a solid of revolution, specifically using the disk method. The equation $y = \sqrt{49 - x^2}$ represents a semicircle with a radius of 7 (since $49 = 7^2$). When this semicircle is revolved around the x-axis, it generates a sphere with radius 7. The volume of a sphere is given by $\frac{4}{3}\pi r^3$, substituting $r = 7$ gives $\frac{4}{3}\pi (7)^3 = \frac{4}{3}\pi (343) = \frac{1372}{3}\pi$. However, the correct formula to apply here is directly related to the area under the curve being revolved, and the error in the explanation led to the incorrect choice. The correct calculation for the volume of the solid generated by revolving the semicircle around the x-axis should indeed consider the entire solid shape formed, which is a hemisphere, not a full sphere, and the correct approach involves integrating the square of the function over the given interval. The correct volume is $\pi \int_{0}^{7} (\sqrt{49 - x^2})^2 dx = \pi \int_{0}^{7} (49 - x^2) dx$, which evaluates to $\frac{686}{3}\pi$, making the correct answer $\frac{686}{3}\pi$, not the volume of a full sphere.

Solve the Problem xxx -Axis the Region Bounded by the Following Graph

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