Calculating volumes of solids is a key application of integration in calculus. We'll look at methods like cross-sectional integration, the , and the to find volumes of various shapes.
These techniques involve slicing solids into thin sections, finding the area of each slice, and integrating to sum up the total volume. We'll see how to set up and solve these integrals for different types of solids.
Volumes of Solids
Volumes by cross-sectional integration
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Determining Volumes by Slicing · Calculus View original
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Determining Volumes by Slicing · Calculus View original
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Top images from around the web for Volumes by cross-sectional integration
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finds volume of solid by integrating area of cross-sections
Select axis to slice solid along (x or y-axis)
Identify shape at generic point on chosen axis (rectangle, circle, triangle)
Express cross-section area as function of chosen variable
Integrate area function over appropriate interval to calculate volume (limits based on solid's boundaries)
Volume formula with slicing method: V=∫abA(x)dx or V=∫cdA(y)dy
A(x) or A(y) cross-section area at given point
a, b, c, d integration limits determined by solid's boundaries
Examples:
Rectangular prism sliced along x-axis has constant rectangular cross-sections
Cone sliced perpendicular to its axis has circular cross-sections with varying radii
The cross-sectional area is a key concept in this method, representing the area of the slice perpendicular to the axis of integration
Disk method for revolution solids
Disk method calculates volume of solid formed by revolving region around horizontal or vertical axis
Determine region bounded by function(s) and axis of revolution
Find disk radius at generic point on revolution axis
Express cross-sectional area using circle area formula (A=πr2)
Integrate area function over appropriate interval for volume (limits based on region's boundaries)
Volume formulas with disk method:
Revolution around x-axis: V=π∫ab[f(x)]2dx
Revolution around y-axis: V=π∫cd[g(y)]2dy
f(x) or g(y) function(s) bounding region
a, b, c, d integration limits determined by region's boundaries
Examples:
Revolving region under parabola y=x2 from x=0 to x=1 around x-axis
Revolving region between x=y2 and x=2 around y-axis
The axis of rotation is crucial in determining the shape of the resulting solid
Washer method for hollow solids
Washer method finds volume of solid generated by revolving region between two functions around horizontal or vertical axis
Identify region bounded by two functions and revolution axis
Determine washer's outer and inner radii at generic point on revolution axis
Subtract inner circle area from outer circle area for cross-sectional area
Integrate area function over appropriate interval to calculate volume (limits based on region's boundaries)
Volume formulas with washer method:
Revolution around x-axis: V=π∫ab([f(x)]2−[g(x)]2)dx
Revolution around y-axis: V=π∫cd([f(y)]2−[g(y)]2)dy
f(x), g(x) or f(y), g(y) outer and inner functions bounding region
a, b, c, d integration limits determined by region's boundaries
Examples:
Revolving region between y=x and y=x2 from x=0 to x=1 around x-axis
Revolving region between x=y2 and x=4y2 from y=1 to y=2 around y-axis
Additional Volume Calculation Methods
Cylindrical shells method: an alternative to disk and washer methods for revolving solids
Involves integrating the volume of thin cylindrical shells
Useful when integrating with respect to the variable perpendicular to the axis of rotation
The definite integral is used in all these methods to sum up infinitesimal volumes
The plane region being revolved determines the limits of integration and the functions used in the volume formula