Technique for calculating volume of solid of revolution

The washer method is a technique in integral calculus for determining the volume of a solid of revolution

Refinement of the disk method

Scenario where solid is generated by rotating a region around an axis that does not coincide with the region's edge

The washer method refines the disk method for scenarios where the solid is generated by rotating a region around an axis that does not coincide with the region's edge

Derived from the cross-sectional shape of the solid

The term "washer method" is derived from the cross-sectional shape of the solid, which resembles a washer - a disk with a central hole

Composition of numerous thin, disk-like washers

The washer method conceptualizes the solid of revolution as being composed of numerous thin, disk-like washers

Calculation of area and volume of individual washer

The area of a single washer is the difference between the areas of two concentric circles, and its volume is found by multiplying its area by its thickness

Transition from discrete summation to continuous integration

For an accurate volume calculation, the washer method transitions from discrete summation to continuous integration by considering the washers to be infinitesimally thin

Integral representation of the aggregate volume of the solid

The aggregate volume of the solid is represented by the integral of the area of the washer cross-section over the specified interval

Calculation of volume for different axis of rotation

Revolving around the x-axis

When revolving around the x-axis, the volume is calculated by integrating the difference between the squares of the radii functions over the specified interval

Revolving around the y-axis

When revolving around the y-axis, the volume is calculated by integrating the difference between the squares of the radii functions over the adjusted interval

Utilizing the Washer Method

To utilize the washer method, one must sketch the region, identify the axis of rotation and boundaries, and integrate the difference between the squares of the radii functions over the specified interval

Example 1: Functions \( f(x) = -x^2 + 3 \) and \( g(x) = -x^2 + 2 \) on the interval \( [0, 1] \)

Revolving these functions around the x-axis and applying the washer method yields the volume \( \frac{13\pi}{3} \)

Example 2: Functions \( p(x) = e^x \) and \( q(x) = 1 \) on the interval \( [0, 2] \)

Revolving these functions around the x-axis and applying the washer method yields the volume \( \frac{\pi}{2}(e^4-5) \)

Applications of the Washer Method

The washer method is an invaluable calculus technique for determining the volume of solids of revolution with hollow interiors, extending the disk method to accommodate regions not bounded by the axis of rotation