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Washer Method: Area of Annular Region

Washer Method: Area of Annular Region
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2018-11-22 10:31
CalculusIntegralGeometric Construction
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources
This example demonstrates how to calculate the area of an annular region using the Washer Method, a key concept in integral calculus. It involves functions, geometric transformations, and symmetry. Constructed with GeoGebra commands like Polygon, Surface, Sequence, Rotate, and Textfield, it is suitable for high school and above students learning about applications of integration in calculus.

GeoGebra Commands & Parameters

  1. Polygon(A, B, D, C) → q1, a_1, b_1, d, c
  2. Surface(x(B), ((u * y(D)) * cos(v)), ((u * y(D)) * sin(v)), u, 0, 1, v, 0, α) → f
  3. Surface(x(A), ((u * y(D)) * cos(v)), ((u * y(D)) * sin(v)), u, 0, 1, v, 0, α) → g
  4. Sequence(Surface[a + (i * deltax) + (deltax * u), (h(a + ((i + 1) * deltax)) * cos(v)), (h(a + ((i + 1) * deltax)) * sin(v)), u, 0, 1, v, 0, α], i, 0, n - 1, 1) → L_4
  5. Sequence(Polygon[(a + (i * deltax), p(a + ((i + 1) * deltax))), (a + ((i + 1) * deltax), p(a + ((i + 1) * deltax))), (a + ((i + 1) * deltax), h(a + ((i + 1) * deltax))), (a + (i * deltax), h(a + ((i + 1) * deltax)))], i, 0, n - 1) → diffrectangles
  6. Sequence(Surface[a + (i * deltax) + (deltax * u), (p(a + ((i + 1) * deltax)) * cos(v)), (p(a + ((i + 1) * deltax)) * sin(v)), u, 0, 1, v, 0, α], i, 0, n - 1, 1) → l1
  7. Rotate(diffrectangles, α, xAxis) → diffrectangles'
  8. Textfield(h) → InputBox1
  9. Textfield(p) → InputBox2
  10. Textfield(a) → InputBox3
  11. Textfield(b) → InputBox4
  12. Textfield(n) → InputBox5
  13. Sequence(Surface[a + (i * deltax), (cos(v) * ((h(a + (i * deltax)) * u) + (p(a + (i * deltax)) * (1 - u)))), (sin(v) * ((h(a + (i * deltax)) * u) + (p(a + (i * deltax)) * (1 - u)))), u, 0, 1, v, 0, α], i, 1, n, 1) → L_2
  14. Sequence(Surface[a + ((i - 1) * deltax), (cos(v) * ((h(a + (i * deltax)) * u) + (p(a + (i * deltax)) * (1 - u)))), (sin(v) * ((h(a + (i * deltax)) * u) + (p(a + (i * deltax)) * (1 - u)))), u, 0, 1, v, 0, α], i, 1, n, 1) → L_3