Math Problem Statement

Do question 27

Do question 27: Find the volume of a cone of height h and base radius r by integration.

Solution

Question 27 asks to "find the volume of a cone of height hh and base radius rr by integration." Let me explain how to solve this.


Solution:

The volume of the cone can be derived using the concept of integration. A cone is a solid of revolution generated by rotating a line segment (forming the side of the cone) around the xx-axis.

  1. Equation of the Line:
    If the cone has height hh and radius rr, we can place the cone in a coordinate system where:

    • The vertex is at (0,0)(0, 0),
    • The base is at x=hx = h,
    • The slope of the line defining the side of the cone is rh-\frac{r}{h}.

    Thus, the equation of the line is: y=rhxy = \frac{r}{h}x

  2. Volume Formula for Solids of Revolution:
    The volume of a solid of revolution about the xx-axis is given by: V=πx=0x=h[y(x)]2dxV = \pi \int_{x=0}^{x=h} [y(x)]^2 \, dx

  3. Substitute y(x)y(x):
    Substitute y=rhxy = \frac{r}{h}x into the formula: V=π0h(rhx)2dxV = \pi \int_{0}^{h} \left( \frac{r}{h}x \right)^2 \, dx

  4. Simplify the Integral:
    Expand and simplify: V=π0hr2h2x2dxV = \pi \int_{0}^{h} \frac{r^2}{h^2}x^2 \, dx V=πr2h20hx2dxV = \frac{\pi r^2}{h^2} \int_{0}^{h} x^2 \, dx

    The integral of x2x^2 is: x2dx=x33\int x^2 \, dx = \frac{x^3}{3}

  5. Evaluate the Definite Integral:
    Evaluate from x=0x = 0 to x=hx = h: V=πr2h2[x33]0hV = \frac{\pi r^2}{h^2} \left[ \frac{x^3}{3} \right]_{0}^{h} V=πr2h2h33V = \frac{\pi r^2}{h^2} \cdot \frac{h^3}{3}

  6. Simplify the Expression:
    Cancel h2h^2 and simplify: V=πr2h3V = \frac{\pi r^2 h}{3}


Final Answer:

The volume of the cone is: V=πr2h3V = \frac{\pi r^2 h}{3}


Would you like further clarification or related problems? Here are some questions to explore further:

  1. How do you find the volume of a sphere using integration?
  2. Can you find the surface area of a cone using integration?
  3. How is the formula for the volume of a cone different from that of a cylinder?
  4. What happens to the volume of the cone if the height is doubled?
  5. Can you calculate the volume for a cone with specific values of rr and hh?

Tip: Always sketch the problem to understand the geometry better before setting up the integral!

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Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Volumes of Revolution

Formulas

Volume of a solid of revolution: V = π∫[y(x)]² dx
Equation of a line: y = mx + b
Integral of x^n: ∫x^n dx = (x^(n+1))/(n+1)

Theorems

Fundamental Theorem of Calculus
Properties of Definite Integrals

Suitable Grade Level

Grades 11-12, Undergraduate