Question

Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.

Accepted Answer

Identify the region bounded by the curves: the upper curve is given by $$y = \sin x + \sec x$$, the lower boundary is $$y = 0$$, and the vertical boundaries are $$x = 0$$ and $$x = \frac{\pi}{3}. $$Since the solid is formed by revolving this region about the x-axis, use the disk method to find the volume. The volume element is a disk with radius equal to the function value $$y at $$each $$x$$, so the volume is given by integrating the area of these disks along the x-axis. Write the volume integral as $$V = \pi \int_{0}^{\frac{\pi}{3}} \left( \sin x + \sec x \$$right)^2$$ \, dx$$, where the integrand is the square of the radius function. Expand the integrand $$\left( \sin x + \sec x \$$right)^2$$ to simplify the integral before integrating. This will give $$\$$sin^2 x + 2$$ \sin x \sec x + \$$sec^2 x$. Set up the integral with the expanded terms and prepare to integrate each term separately over the interval $$[0, \frac{\pi}{3}]$$. Remember to use appropriate trigonometric identities and integration techniques for each term.

Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.

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Key Concepts

Volume of Solids of Revolution

Disk Method

Integration of Trigonometric and Secant Functions