Calculus 1 Lecture 5.2: Volume of Solids By Disks and Washers Method

Professor Leonard・2 minutes read

Understanding the volume of figures involves cutting shapes into slabs to determine the cross-sectional area, using integration to sum up the volumes of these slabs accurately. The process includes finding surface areas of circles when revolving functions around axes and properly identifying top and bottom functions for correct volume calculations.

Insights

  • Understanding the process of slicing volumes into slabs and finding the cross-sectional area is crucial for determining the volume of a solid figure through integration.
  • The integration method for finding the volume of a solid bound by planes perpendicular to the x-axis involves summing up volumes of slabs from the first to the nth interval, refining the approximation by taking the limit as n approaches infinity.
  • The concept of revolving shapes around different axes to create unique volumes like cylinders, spheres, and cones is explained, with formulas for calculating their volumes through integration.
  • The importance of identifying top and bottom functions, adjusting variables based on the axis of rotation, and evaluating integrals accurately is emphasized when finding volumes between curves and revolving them around specific axes.

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Recent questions

  • How is the volume of a solid figure determined?

    By finding the cross-sectional area and length.

  • What shapes can be created by revolving figures around an axis?

    Cylinders, spheres, cones, and torus shapes.

  • How is the volume of a sphere calculated through calculus?

    By revolving a half circle around the x-axis.

  • What is the process for finding volumes between curves?

    Setting up integrals and identifying top functions.

  • How are volumes calculated when revolving around different axes?

    Adjusting functions and bounds based on the axis.

Related videos

Summary

00:00

Volume of Figures: Slicing and Integration Method

  • Discussing the application of finding the volume of figures using discs and washers.
  • Explaining the concept of slicing volumes to find the volume of a solid figure.
  • Describing the process of cutting a figure into thin slabs to find the volume.
  • Emphasizing the importance of finding the cross-sectional area to determine volume.
  • Illustrating the method of finding the volume of a solid bound by planes perpendicular to the x-axis.
  • Stating that the volume of a figure is the surface area of the cross-section times its length.
  • Clarifying that the method works for figures with sides perpendicular to the x-axis.
  • Detailing the process of cutting figures into slabs of equal width represented by Delta X.
  • Demonstrating the use of arbitrary points like XK dot within each interval to find volume.
  • Highlighting the similarity between finding volume and the area under a curve in terms of integration.

21:11

Finding Volumes Through Cross-Sectional Integration

  • The width of the cross-sectional area is determined by the distance from point A to point B, while the focus is on finding the cross-sectional area at an arbitrary point XK dot.
  • To find the cross-sectional area, an arbitrary point XK dot is chosen on each sub-interval, and the surface area at that point is calculated.
  • The volume of each interval is represented by the area at XK dot multiplied by the width (Delta X), giving the volume of one slab.
  • To find the total volume, all slabs' volumes are summed up by adding them from the first to the nth interval.
  • Taking the limit as n approaches infinity refines the approximation to an exact volume calculation through integration.
  • The integral from A to B represents the volume of a solid, with A and B being the starting and ending points perpendicular to the x-axis.
  • The same process can be applied along the y-axis, with C and D as the starting and ending points perpendicular to the y-axis.
  • The volume of a solid is found by integrating the cross-sectional area from A to B or C to D, depending on the axis of revolution.
  • A simple example involves a uniform function where the cross-sectional area is a constant (Pi) and the volume is calculated by integrating from the start to end points.
  • Solid of Revolution is created by revolving a shape around an axis, resulting in unique volumes like cylinders or spheres, calculated through integration.

41:46

"Shapes Formed by Revolving Functions"

  • When revolving a full circle, it creates the same shape, possibly with twice the volume due to additional sides.
  • Revolving a half circle results in a sphere.
  • Sweeping a figure around can produce a cone.
  • A donut shape, or torus, is formed when a figure with a hole is swept around.
  • A cylinder within a cylinder is created by sweeping a figure with a hole in the middle.
  • Standard formulas exist for calculating the volumes of shapes like cylinders, spheres, and cones.
  • Sweeping a favorite function from A to B can create a unique shape with curved edges.
  • The sides of the solid created by rotating a function around the x-axis are perpendicular to the x-axis.
  • The volume of three-dimensional objects can be calculated by integrating the surface area from A to B.
  • The surface area of the cross-section when revolving a function around the x-axis is a circle, with the radius equal to the function's height.

01:02:56

Calculating Volumes Using Integrals and Functions

  • The volume formula involves an integral from A to B of Pi fx^2 DX.
  • The function f(x) is 3√x, leading to 3√x being squared.
  • The integral is evaluated from 1 to 4, resulting in 9x.
  • Simplifying the integral gives 9Ο€ times the integral from 1 to 4 of x DX.
  • An alternative method involves pulling out constants before evaluation.
  • The final result of the integral calculation is 15.
  • The volume of a shape rotated around the x-axis is determined by finding the height of the function at each point.
  • To derive the volume of a sphere, a half circle is revolved around the x-axis.
  • The function for the half circle is determined by solving for y in the equation x^2 + y^2 = radius^2.
  • The volume of the sphere is calculated using the integral from -R to R of Pi times the function squared DX.

01:20:41

Volume Calculations Using Integration and Rotation

  • The integral involves negative R cubed and positive RB cubed, resulting in a simplified expression.
  • The expression is further simplified by combining like terms and finding a common denominator.
  • The process involves multiplying by 3 over 3 to simplify the expression.
  • The final expression is found to be 6RB over 3 minus 2RB over 3, with a common denominator.
  • The concept of finding the volume of a sphere through calculus is discussed, emphasizing the use of small circles to calculate the volume.
  • The method of finding volumes of solids rotated around an axis by summing up the surface areas of circles is explained.
  • The idea of a solid with a hole in the middle, resembling a vase, is introduced, highlighting the need to find the cross-sectional area for integration.
  • The process of finding the volume of a solid created by rotating the area between two functions around the x-axis is detailed.
  • The importance of determining which function is on top for correct subtraction and positive volume calculation is emphasized.
  • The method of determining which function is on top by plugging in a value within the interval is explained, simplifying the setup for finding the volume.

01:40:28

Volume of Solids: Calculations and Formulas

  • The formula for finding the volume of a solid revolved around the x-axis involves subtracting the volumes of two discs, outer and inner, squared due to radius considerations.
  • The process involves integrating the difference between two functions, F(x) and G(x), to find the volume of the solid.
  • An alternative method to derive the formula involves finding the volumes of F(x) and G(x) separately and then subtracting them.
  • Simplifying the integral of 1/4 + x^4 from 0 to 2 results in 69Ο€ as the volume of the solid.
  • The volume calculated represents the amount of concrete needed if the solid were to be poured into a mold.
  • When revolving a figure around the y-axis, the formulas and variables used differ from those when revolving around the x-axis.
  • For volumes perpendicular to the y-axis, the area function is represented as Pi*r^2*dy, with functions and bounds in terms of y.
  • A practical example involves finding the volume of a solid formed by revolving y=x around the y-axis between y=0 and y=2, resulting in a volume of 32Ο€/5.
  • To revolve a figure around the y-axis, the function must be in terms of y, requiring solving for x to ensure correct calculations.
  • The process of finding volumes when revolving around different axes involves adjusting functions and bounds accordingly to achieve accurate results.

02:01:17

"Calculus: Finding Volumes with Variable Substitution"

  • The problem involves finding the value of X when Y is known to be 2, resembling a u substitution concept.
  • By substituting 2 for Y, the solutions for X are found to be 4 and 0.
  • The integral from 0 to 4 is calculated, resulting in a straightforward Pi integral of X DX.
  • The integral simplifies to Pi times X^2 over 2 from 0 to 4, yielding a difference of Pi 32.
  • The discussion emphasizes the importance of understanding variables and functions in calculus problems.
  • Different methods for finding volumes, such as cylindrical shell, are introduced, highlighting the need to switch variables based on the axis of rotation.
  • Practical advice is given to ensure a clear understanding of which variables to use based on the axis of rotation.
  • The text transitions to a practice stage, focusing on finding volumes between curves and revolving them around specific axes.
  • Detailed instructions are provided on setting up integrals for volume calculations, emphasizing the importance of identifying top and bottom functions.
  • The process of solving for volumes between curves and revolving them around different axes is explained, with practical examples and step-by-step guidance.

02:18:16

"Revolve around Y-axis to find volume"

  • To solve the problem, you need to factor y = ybus 1.
  • Solving the problem leads to y = 0 and Y Cub - 1 equal 0.
  • The only way to get y = 1 is if yal 1, not negative 1.
  • When solving for y, the cube root of one is one.
  • Integral starts and stops along the y-axis, from 0 to 1 in terms of Y.
  • The concept of top and bottom changes when revolving around the Y-axis.
  • To determine which function is on top, plug in numbers to compare.
  • The integral formula involves Pi, square root of Y, and Y squared.
  • Evaluating the integral involves plugging in 1 and 0 to get the final result.
  • Finding the volume of a solid revolves around understanding the concept of revolving around a specific line, like y = 3.

02:37:54

Calculating Surface Area and Integrating Functions

  • To find the surface area of the outside Circle minus the inside Circle, square the radii, multiply by pi, and subtract them.
  • Moving the point of Revolution changes the area as the object sweeps out more space when further away from the x-axis.
  • The formula for the surface area involves squaring the radii of the inside and outside circles, subtracting them, and multiplying by pi.
  • When integrating, start by dealing with roots and combining like terms before proceeding to save time and effort.
  • After simplifying the integral, plug in the values (0 and 1) to find the final result, ensuring correct evaluation to avoid errors.
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