- Find the volume of a solid by using the slicing method
- Find the volume of a solid by using the disk method
- Compute the volume of a hollow solid of revolution by using the washer technique
Volume and the Slicing Method
The Main Idea
- Volume is a measure of three-dimensional space occupied by a solid. While we have formulas for basic shapes (e.g., V=43πr3V=43πr3 for a sphere), not all solids have simple formulas.
- The slicing method is a technique for finding volumes of solids with varying cross-sections, extending the concept of definite integrals to three dimensions.
- Key formula:
- V=∫baA(x)dxV=∫baA(x)dx, where A(x)A(x) is the cross-sectional area
- A cylinder, in mathematical terms, is any solid generated by translating a plane region along a line perpendicular to it (the axis). This generalizes the common notion of a circular cylinder.
- Cross-sections are two-dimensional slices of a three-dimensional solid, obtained by intersecting the solid with a plane. The shape and area of these cross-sections are crucial in applying the slicing method.
- The choice of slicing direction (e.g., perpendicular to xx-axis, yy-axis, or zz-axis) can significantly affect the complexity of calculations. Selecting the appropriate direction is a key problem-solving skill.
Slicing Method Process
- Divide the solid into thin slices perpendicular to a chosen axis
- Estimate volume of each slice: V(Si)≈A(x∗i)ΔxV(Si)≈A(x∗i)Δx
- Sum the volumes of all slices: V(S)≈∑ni=1A(x∗i)ΔxV(S)≈∑ni=1A(x∗i)Δx
- Take the limit asnn approaches infinity to get the definite integral
Problem-Solving Strategy
- Determine the shape of a cross-section of the solid
- Find a formula for the area of the cross-section
- Integrate the area formula over the appropriate interval
Use the slicing method to derive the formula V=13πr2hV=13πr2h for the volume of a circular cone.
Find the volume of a right circular cone with radius rr and height hh using the slicing method.
Solids of Revolution and the Slicing Method
The Main Idea
- A solid of revolution is formed by rotating a planar region around a line in that plane
- Cross-sections of these solids perpendicular to the axis of revolution are typically circular
- Volume formula:
- V=∫baA(x)dxV=∫baA(x)dx, where A(x)A(x) is the area of the circular cross-section
- For rotation around xx-axis:
- A(x)=π[f(x)]2A(x)=π[f(x)]2, where f(x)f(x) is the function being rotated
- The choice of axis of revolution affects the complexity of the integration
- Solids of revolution are common in manufacturing and engineering applications
Problem-Solving Strategy
- Sketch the region being rotated and visualize the resulting solid
- Identify the axis of rotation and the limits of integration
- Determine the formula for the cross-sectional area A(x)A(x)
- Set up and evaluate the integral V=∫baA(x)dxV=∫baA(x)dx
Find the volume of the solid formed by rotating y=4−x2y=4−x2 from x=0x=0 to x=2x=2 around the xx-axis.
Find the volume of the solid formed by rotating y=x+1y=x+1 from x=1x=1 to x=3x=3 around the xx-axis.
Find the volume of the solid formed by rotating y=1xy=1x from x=1x=1 to x=∞x=∞ around the xx-axis.
The Disk Method
The Main Idea
- The Disk Method is a specialized application of the slicing method for solids of revolution
- It uses circular disks to approximate the volume of the solid
- Key formula for rotation around xx-axis:
- V=∫baπ[f(x)]2dxV=∫baπ[f(x)]2dx
- Key formula for rotation around yy-axis:
- V=∫dcπ[g(y)]2dyV=∫dcπ[g(y)]2dy
- Applies to solids formed by rotating a region bounded by a function and an axis
- Can be used for rotation around any horizontal or vertical line, not just coordinate axes
Problem-Solving Strategy
- Identify the function and the axis of rotation
- Determine the limits of integration (the interval over which the region is rotated)
- Set up the integral using the appropriate formula
- Evaluate the integral to find the volume
The Washer Method
The Main Idea
- The Washer Method is used for solids of revolution with cavities
- Applies when the region being revolved is bounded by two functions or when the axis of revolution is not the boundary of the region
- Key formula for rotation around xx-axis:
- V=∫baπ[(f(x))2−(g(x))2]dxV=∫baπ[(f(x))2−(g(x))2]dx
- Key formula for rotation around yy-axis:
- V=∫dcπ[(u(y))2−(v(y))2]dyV=∫dcπ[(u(y))2−(v(y))2]dy
- The cross-sectional area is the difference between the areas of two circles
- Can be applied to rotation around any horizontal or vertical line, not just coordinate axes
Problem-Solving Strategy
- Identify the functions bounding the region and the axis of rotation
- Determine the radii of the outer and inner circles at each cross-section
- Set up the integral using the appropriate washer formula
- Evaluate the integral to find the volume
Find the volume of a solid of revolution formed by revolving the region bounded above by the graph of f(x)=x+2f(x)=x+2 and below by the x-axisx-axis over the interval [0,3][0,3] around the line y=−1.y=−1.
60π60π units3