Question

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.

Accepted Answer

Identify the region bounded by the curve $$f(x) = \sin(x)$$ and the x-axis on the interval $$[0, \pi]. $$This region lies between the curve and the x-axis from $$x=0 to$$ $$x=\pi. $$Since the solid is generated by revolving this region about the y-axis, consider using the method of cylindrical shells. The formula for the volume using cylindrical shells is:

$$V = \int_a^b 2\pi \cdot (	ext{radius}) \cdot (	ext{height}) \, dx In $$this problem, the radius of a shell at position $$x is $$the distance from the y-axis, which is $$x$$, and the height of the shell is the function value $$\sin(x). So $$the volume integral becomes:

$$V = \$$int_0$$^{\pi} 2\pi x \sin(x) \, dx$$ Set up the integral $$\$$int_0$$^{\pi} 2\pi x \sin(x) \, dx$$ and prepare to evaluate it. You can factor out the constant $$2\pi to $$simplify the integral to $$2\pi \$$int_0$$^{\pi} x \sin(x) \, dx. To $$evaluate $$\$$int_0$$^{\pi} x \sin(x) \, dx$$, use integration by parts where you let $$u = x (so$$ $$du = dx) $$and $$dv = \sin(x) dx (so$$ $$v = -\cos(x)). $$Apply the integration by parts formula:

$$\int u \, dv = uv - \int v \, du.$$

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Volume of Solids of Revolution

Shell Method

Properties of the Sine Function on [0, π]

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Volume of Solids of Revolution

Shell Method

Properties of the Sine Function on [0, π]

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.