Solids of Revolutions Using Definite Integrals

Solids of Revolutions Using Definite Integrals:

  1. Objective:

    • Calculate volumes of solids formed by rotating a curve around a specific axis using definite integrals.
  2. Approach:

    • The method involves revolving a function y=f(x) around the x-axis or y-axis to form a solid.
    • Disk method: Rotating about the x-axis.
    • Washer method: Rotating between two functions about the x-axis.
    • Shell method: Rotating about the y-axis.
  3. Disk Method (Rotating Around x-axis):

    • Formula: The volume of the solid formed by rotating the curve y=f(x) over [a,b] around the x-axis is given by: V=πab[f(x)]2dx
  4. Washer Method (Rotating Between Curves Around x-axis):

    • Formula: The volume of the solid between two curves y=f(x) and y=g(x) over [a,b] rotated around the x-axis is given by: V=πab[(f(x))2(g(x))2]dx
  5. Shell Method (Rotating Around y-axis):

    • Formula: The volume of the solid formed by rotating the curve x=f(y) over [c,d] around the y-axis is given by: V=2πcdxf(y)dy
  6. Steps to Find Volume:

    • Step 1: Identify Rotation Axis: Determine whether the rotation is around the x-axis or y-axis.

    • Step 2: Set Up Integral: Based on the method (disk, washer, or shell), set up the appropriate definite integral formula to calculate volume.

    • Step 3: Define Limits: Define the limits of integration (a, b, c, d) corresponding to the interval of the curve or region being rotated.

    • Step 4: Evaluate Integral: Compute the definite integral to find the volume of the solid of revolution.

  7. Applications:

    • Engineering: Used to calculate volumes of components, such as pipes, tanks, and machine parts.

    • Physics: Applied to find volumes of rotated shapes in thermodynamics, fluid dynamics, and mechanics.

Example: Disk Method for Solid of Revolution

Consider the curve y=x2 over the interval [0,2]. Let's find the volume of the solid formed by rotating this curve around the x-axis.

  1. Objective:

    • Calculate the volume of the solid generated by revolving y=x2 over [0,2] around the x-axis using the disk method.
  2. Disk Method Formula:

    • The formula for volume using the disk method is: V=πab[f(x)]2dx
  3. Steps to Find Volume:

    • Step 1: Identify Rotation Axis: The rotation is around the x-axis.

    • Step 2: Setup Integral for Disk Method:

      • The formula for the disk method: V=πab[f(x)]2dx.
      • For y=x2 over [0,2], the integral becomes: V=π02(x2)2dx=π02x4dx
    • Step 3: Evaluate Integral:

      • Compute the definite integral: V=π[x55]02=π(255055)=π325=32π5
  4. Result:

    • The volume of the solid generated by rotating the curve y=x2 over [0,2] around the x-axis using the disk method is 32π5 cubic units.