"Determine whether the following statements are true and give an explanation or counterexample.


b. The volume of a hemisphere can be computed using the disk method. "

Verified step by step guidance

1. Step 1: Recall the disk method, which is used to calculate the volume of a solid of revolution. The formula involves integrating the area of circular disks perpendicular to the axis of rotation. The general formula is: $V\; =\; \int [a,b]\; \pi r²\; dx$, where r is the radius of the disk at a given x-coordinate.
2. Step 2: Visualize the hemisphere as a solid of revolution. A hemisphere can be generated by rotating a semicircle about its diameter (e.g., the x-axis). The equation of a semicircle with radius R centered at the origin is $y\; =\; \surd (R²\; -\; x²)$.
3. Step 3: Set up the integral using the disk method. The radius of each disk is given by the y-coordinate of the semicircle, $r\; =\; \surd (R²\; -\; x²)$. The volume of the hemisphere is then computed as: $V\; =\; \int [-R,R]\; \pi (\surd (R²\; -\; x²))²\; dx$.
4. Step 4: Simplify the integrand. Squaring the radius gives $(\surd (R²\; -\; x²))²\; =\; R²\; -\; x²$. The integral becomes: $V\; =\; \pi \int [-R,R]\; (R²\; -\; x²)\; dx$.
5. Step 5: Evaluate the integral. Break it into two parts: $\pi \int [-R,R]\; R²\; dx$ and $-\pi \int [-R,R]\; x²\; dx$. Compute each term separately to find the volume of the hemisphere. This confirms that the disk method can indeed be used to compute the volume of a hemisphere.

Key Concepts

Disk Method

The disk method is a technique used in calculus to find the volume of a solid of revolution. It involves slicing the solid into thin disks perpendicular to an axis of rotation, calculating the area of each disk, and then integrating these areas over the interval of interest. This method is particularly useful for solids generated by rotating a function around a horizontal or vertical axis.

Volume of a Hemisphere

The volume of a hemisphere is calculated using the formula V = (2/3)πr³, where r is the radius of the hemisphere. This formula derives from the volume of a full sphere, V = (4/3)πr³, halved to account for the hemisphere. Understanding this volume is essential when applying the disk method, as it provides a benchmark for verifying the correctness of the computed volume.

Integration

Integration is a fundamental concept in calculus that involves finding the accumulated area under a curve or the total volume of a solid. In the context of the disk method, integration is used to sum the volumes of all the infinitesimally thin disks. Mastery of integration techniques, such as definite and indefinite integrals, is crucial for accurately applying the disk method to compute volumes.