Name: finding sphere volume with spherical wedges
Uploaded: 2026-06-08T09:38:36.500Z
Duration: 1 min 27 s
Channel: JoVE

Triple integrals in spherical coordinates are ideal for objects with central symmetry.In spherical coordinates, any point in space is described using three variables.Here, rho represents the distance from the origin, while theta is the angle in the xy plane, measured from the x-axis, and phi is the angle measured from the positive z-axis.These coordinates can be seamlessly used to derive the standard formula for the volume of a sphere. First, the sphere is divided into tiny spherical wedges, each defined by changes in rho, theta, and phi.The volume of each wedge is found by multiplying its radial thickness by its two distinct circular arc lengths.The total volume is then calculated by integrating this small wedge over the entire sphere.To set the limits, rho ranges from 0 at the center to the outer radius R. Next, phi sweeps vertically from 0 to pi.Finally, theta rotates a full 0 to 2pi around the z-axis to sweep out the entire three-dimensional shape.Solving this integral with these limits gives the known formula for the volume of a sphere of radius R using spherical coordinates.

triple integrals in spherical coordinates

Triple integrals in spherical coordinates provide an efficient method for evaluating volumes over regions with central symmetry, such as spheres. Instead of describing points by rectangular coordinates, spherical coordinates use three variables: &#x1D70C;, &#x1D703;, and &#x1D711;. Here, &#x1D70C; is the distance from the origin, &#x1D703; is the angle in the xy-plane measured from the positive x-axis, and &#x1D711; is the angle measured downward from the positive z-axis.To derive the volume of a sphere, the solid region can be divided into small spherical wedges. Each wedge is determined by small changes in &#x1D70C;, &#x1D703;, and &#x1D711;. Its approximate volume is found by multiplying the radial thickness by two circular arc lengths. This wedge structure accounts for how distances spread out as &#x1D70C; increases and as the direction changes over the sphere.For a sphere of radius R, &#x1D70C; ranges from the center of the sphere to its outer surface. The angle &#x1D711; sweeps vertically from the positive z-axis to the negative z-axis, covering the full height of the sphere. The angle &#x1D703; rotates once around the z-axis, sweeping through the entire circular direction in the xy-plane.The total volume is obtained by adding the volumes of all the tiny spherical wedges throughout the sphere. Because the limits cover every point inside the sphere exactly once, this triple integral gives the full volume of the solid. After evaluation, the result is the standard volume formula for a sphere of radius R.

Triple Integrals in Spherical Coordinates

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JoVE Core

Calculus

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Multiple Integrals and Applications

KEY TERMS AND CONCEPTS

finding sphere volume with spherical wedges

Triple integrals in spherical coordinates are a useful way to find volumes with central symmetry, especially spheres. Spherical coordinates describe a point with three variables: ρ, θ, and φ. Here, ρ is the distance from the origin. θ is the angle in the xy-plane measured from the positive x-axis, and φ is the angle measured downward from the positive z-axis.

To find the volume of a sphere, the solid can be split into many small spherical wedges. Each wedge is defined by small changes in ρ, θ, and φ. Its approximate volume comes from multiplying the radial thickness by two circular arc lengths. This wedge shape shows how distances spread out as ρ gets larger and as direction changes across the sphere.

For a sphere of radius R, ρ extends from the center to the outer surface. The angle φ sweeps vertically from the positive z-axis to the negative z-axis, so it covers the full height of the sphere. The angle θ rotates once around the z-axis and covers the full circular direction in the xy-plane. Adding the volumes of all these tiny wedges gives the total volume. Because the limits include every point inside the sphere exactly once, the triple integral produces the full volume of the solid. The final result matches the standard volume formula for a sphere of radius R.

Finding Sphere Volume with Spherical Wedges

finding-sphere-volume-with-spherical-wedges

introduction to double integrals

Double Integrals for Volume and Density

Double integrals help find the volume under a surface and over a region in the xy-plane. They extend single-variable integration, which finds area under a curve on an interval, to functions of two variables. This makes them useful for models in mathematics, physics, and engineering that need three-dimensional thinking. Single-variable integration uses thin rectangles to approximate area under y = f(x) from a to b. The definite integral is written as a limit of a sum of these rectangle areas.

the midpoint rule for double integrals

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The midpoint rule estimates volume for a double integral over a rectangular region. It is useful when a surface has a continuously changing height. In civil engineering, it can help estimate how much soil must be moved when planning a road across uneven terrain. The road footprint is treated as a rectangle in the xy-plane. The terrain above a flat reference level is described by a continuous height function, f(x, y). The goal is to estimate the volume of soil between that surface and the...

iterated integrals and fubini's theorem

Double Integrals as Cross-Sections

Double integrals help find the volume under a surface z = f(x,y) over a region in the plane. They extend the idea of a single-variable integral to functions of two variables. For a rectangular region with a ≤ x ≤ b and c ≤ y ≤ d, and for functions that are continuous on that region, the double integral can be written as an iterated integral. This breaks one harder problem into two simpler steps. A useful way to understand this method is through cross-sectional area. If x is held constant, then...

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double integrals over general regions

Double Integrals on Curved Boundaries

Double integrals can measure quantities spread across a two-dimensional region, such as rainfall over a lake, heat on a metal plate, or population density over land. In many real situations, the region does not have straight sides. It may have curved or irregular edges instead. That makes the region harder to describe as a simple rectangle. To handle an irregular region D, it is placed inside a larger rectangular region R. A function f(x,y) gives the quantity on D, such as rainfall intensity...

changing the order of integration in double integrals

changing-the-order-of-integration-in-double-integrals

Finding Volume with Double Integral Bounds

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double integrals in polar coordinates

Polar Area Elements for Double Integrals

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triple integrals in rectangular coordinates

triple-integrals-in-rectangular-coordinates

Triple Integrals for Volume and Mass

Triple integrals help find the total value of a function across a three-dimensional region. They are often used to calculate volume, mass, and other physical quantities that change from place to place. The idea is to split a solid region into many small rectangular boxes, evaluate the function in each box, and add the results. As the boxes get smaller, the triple Riemann sum gets closer to the exact triple integral. In rectangular coordinates, the integral is written as an iterated integral,...

triple integrals over general regions

Evaluating Triple Integrals by Region

Triple integrals over bounded solid regions are used to measure a continuous function across a three-dimensional shape. The solid region E is often placed inside a rectangular box B. To handle the region cleanly, the function f(x,y,z) is extended to a new function F that matches f inside E and is zero outside the solid. The triple integral is written as ∭_E f(x,y,z) dV. For the integral to exist, the function f must be continuous. The boundary of E also needs to be smooth enough. As with...

changing the order of integration in triple integrals

changing-the-order-of-integration-in-triple-integrals

Setting Triple Integral Bounds for Easier Evaluation

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triple integrals in cylindrical coordinates

triple-integrals-in-cylindrical-coordinates

Volume of Cylinders with Triple Integrals

Cylindrical coordinates describe a point in three-dimensional space with three values: radial distance, angle, and height. The height gives the position above the xy-plane. The radial distance tells how far the point is from the z-axis, and the angle gives its direction from the positive x-axis in the xy-plane. This system is useful for regions with circular symmetry. It matches the shape of cylinders, disks, and circular tanks. For that reason, it is a natural choice when finding volume in...

Triple integrals in spherical coordinates are a useful way to find volumes with central symmetry, especially spheres. Spherical coordinates describe a point with three variables: ρ, θ, and φ. Here, ρ is the distance from the origin. θ is the angle in the xy-plane measured from the positive x-axis, and φ is the angle measured downward from the positive z-axis. To find the volume of a sphere, the solid can be split into many small spherical wedges. Each wedge is defined by small changes in ρ, θ,...

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change-of-variables-in-multiple-integrals

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real-life-applications-of-multiple-integrals

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key terms and concepts

Learn how spherical coordinates use ρ, θ, and φ to break a sphere into wedges and set up a triple integral for its volume.

Finding Sphere Volume with Spherical Wedges | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Sphere Volume with Spherical Wedges

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