Question

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated?

Accepted Answer

First, identify the region bounded by the curves $$y = \sec x$$ and $$y = 2 on $$the interval $$0 \leq x \leq \frac{\pi}{3}. $$Since $$\sec x = \frac{1}{\cos x}$$, note that $$\sec x is $$increasing on this interval and $$\sec 0 = 1$$, $$\sec \frac{\pi}{3} = 2$$, so the curves intersect at $$x = \frac{\pi}{3}. $$Since the region is revolved about the x-axis, consider the vertical slices perpendicular to the x-axis. The volume can be found using the washer method because the region lies between two curves above the x-axis. Set up the volume integral using the washer method formula: $$V = \pi \int_a^b \$$left(R(x)^2$$ - r(x)^2$\right) \, dx$$, where $$R(x) is $$the outer radius and $$r(x) is $$the inner radius of the washers. Determine the outer and inner radii. The outer radius $$R(x) is $$the distance from the x-axis to the upper curve $$y=2$$, so $$R(x) = 2. $$The inner radius $$r(x) is $$the distance from the x-axis to the lower curve $$y=\sec x$$, so $$r(x) = \sec x. $$Write the integral explicitly: $$V = \pi \$$int_0$$^{\frac{\pi}{3}} \$$left(2^2$$ - (\sec x)^2$\right) \, dx = \pi \$$int_0$$^{\frac{\pi}{3}} \left(4 - \sec^2$ x\right) \, dx. $$This integral can then be evaluated to find the volume.

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated?

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43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated?

Risposta video verificata per un problema simile:

Concetti chiave

Volume of Solids of Revolution

Disk and Washer Methods

Choosing the Axis and Limits of Integration

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated?