Textbook Question
Suppose a cut is made through a solid object perpendicular to the x-axis at a particular point x. Explain the meaning of A(x).
9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
2:32 minutesProblem 6.3.5c
Textbook Question
Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures).
c. Write an integral for the volume of the solid.
Verified step by step guidance1
Identify the region R bounded by the curve $y = \sqrt{\cos x}$ and the x-axis on the interval $[0, \frac{\pi}{2}]$. This means the region lies between $y = 0$ and $y = \sqrt{\cos x}$ for $x$ in $[0, \frac{\pi}{2}]$.
Since the solid is formed by revolving the region R about the x-axis, use the disk method to find the volume. The volume of a solid of revolution generated by revolving a curve $y = f(x)$ about the x-axis from $x = a$ to $x = b$ is given by the integral $V = \pi \int_a^b [f(x)]^2 \, dx$.
In this problem, the function is $f(x) = \sqrt{\cos x}$, so the radius of each disk is $\sqrt{\cos x}$. Squaring this radius gives the area of the cross-sectional disk as $[\sqrt{\cos x}]^2 = \cos x$.
Set up the integral for the volume using the limits of integration $0$ to $\frac{\pi}{2}$:
$$V = \pi \int_0^{\frac{\pi}{2}} \cos x \, dx.$$
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Region Bounded by a Curve and the x-axis
Understanding the region bounded by the curve y = √cos x and the x-axis on [0, π/2] is essential. This involves identifying the area under the curve from x = 0 to x = π/2, which forms the cross-sectional shape to be revolved around the x-axis.
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Volume of a Solid of Revolution Using the Disk Method
The disk method calculates the volume of a solid formed by revolving a region around the x-axis. The volume is found by integrating π times the square of the radius function (here, y = √cos x) with respect to x over the given interval.
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Setting up Definite Integrals for Volume
Writing the integral requires expressing the volume as an integral with proper limits and integrand. For this problem, the integral is from 0 to π/2 of π times (√cos x)² dx, simplifying to π∫₀^{π/2} cos x dx, which represents the volume of the solid.
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