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Section 3.2 Calculus 2

Volumes by Cross-Sectioning

3.2 Volumes by Cross-Sectioning

3.2.1 Defining Volume with Integrals

  • The volume of a solid defined between \(x=a\) to \(x=b\) with a cross-sectional area of \(A(x)\) at each \(x\)-value is defined to be \(V=\int_a^b A(x)\,dx\).
  • Steps for solving such problems:
    1. Sketch the solid along the \(x\)-axis with a typical cross-section at some \(x\) value.
    2. Find the formula for \(A(x)\), and the minimal/maximal \(x\) values \(a,b\).
    3. Evaluate \(V=\int_a^b A(x)\,dx\).
  • Example Show that the volume of a pyramid with a square base of sidelength \(2\) and height \(3\) is \(4\) cubic units.

3.2.2 Circular Cross-Sections

  • In the case that all cross-sections are circular, we may replace \(A(x)\) with \(\pi[R(x)]^2\), where \(R(x)\) is the radius of the circular cross-section at that \(x\) value.
  • Example Prove that a cone of radius \(r\) and height \(h\) has volume \(V=\frac{1}{3}\pi r^2 h\).

Review Exercises

  1. Find the volume of a solid located between \(x=-1\) and \(x=2\) with cross-sectional area \(A(x)=x^2+1\) for all \(-1\leq x\leq 2\).
  2. Find the volume of a solid located between \(x=0\) and \(x=1\) whose cross-sections are parallelograms with base length \(b(x)=x+1\) and height \(h(x)=x^2+1\) for all \(0\leq x\leq 1\).
  3. Find the volume of a wedge cut from a circular cylinder with radius \(2\), sliced out at a \(45^\circ\) angle from the diameter of its base. (Hint: Sketch the diameter of the cylinder along the \(x\)-axis from \(-2\) to \(2\), and use the equation \(x^2+y^2=2^2\). The cross-sections will be isosceles triangles.)
  4. Prove that the volume of a sphere with radius \(r\) is \(V=\frac{4}{3}\pi r^3\). (Hint: Draw a diameter of the sphere on the \(x\)-axis from \(-r\) to \(r\), and use the equation \(x^2+y^2=r^2\).)
  5. Find the volume of the solid whose base is the region \(0\leq y\leq 4-x^2\) and whose cross-sections are equilateral triangles perpendicular to the \(x\)-axis.


Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 6.1