Question

(1) Al-Kuhi studied the centers of gravity of the following series of shapes. Fix a semicircle ABG with diameter AG, center (of the corresponding circle) D along AG, and BD perpendicular to AG (see figure below). Then form the triangle ABG and the (segment of the) parabola ABG, as pictured. Al-Kuhi considered both these plane figures and the solids obtained by rotating them about the line BD (sweeping out a hemisphere, a cone, and a paraboloid). He announced in a letter to al-Sabi that the centers of gravity of the various planar and solid figures lie along BD at points P dividing BD in the following ratios: (a) Triangle: DP/DB = 1/3. (b) Cone: DP/DB = 1/4. (c) Parabola: DP/DB = 2/5. (d) Paraboloid: DP/DB = 2/6. (e) Semicircle: DP/DB = 3/7. (f) Hemisphere: DP/DB = 3/8. Such calculations are great feats prior to the introduction of calculus, but they are not all correct. Use calculus to derive the correct results for the planar figures (triangle, parabola, semicircle). Here is an outline of one way to proceed: we may assume the circle has radius 1, and that the points A and D are (0, 0) and (1, 0). For any of the figures considered, let f(x) denote the height at x, and let A denote the total area, so that A = ??² f(x)dx. Calculus tells us that the point P is then (1, p) where p = [??² (f(x)²/2) dx] / A Use this formula to compute p for each of the three shapes. For the semicircle, you will find that your answer does not agree exactly with that of al-Kuhi: what value of ? would be implied by the semicircle answer? (Our reading from the Sourcebook, 572-573, is in fact al-Kuhi’s response to a contemporary who questioned the result of his calculation, since it seemed at variance with Archimedes’ approximations to ?. See 568-572 of the Sourcebook for more of this exchange.) (2) This week we will study Ibn al-Haytham’s work on the volume of a paraboloid. Consider a parabola and a line perpendicular to its axis that intersects the parabola in points P and Q and thereby determines a parabolic segment. Consider the solid obtained by rotating the parabolic segment about the line. Also consider the cylinder obtained by rotating the rectangle whose top is PQ and whose bottom is obtained by translating PQ so that it is tangent to the parabola at the vertex. (See picture.) Show that the volume of the paraboloid is 8/15 of the volume of the cylinder in the following steps: (a) Write down an equation of the parabola and an equation of the line through P and Q. (b) Compute the volume of the cylinder (in terms of the parameters from the previous part). (c) Set up and compute an integral that computes the volume of the paraboloid, and compare it to your cylinder calculation.

          (1) Al-Kuhi studied the centers of gravity of the following series of shapes. Fix a semicircle ABG with diameter AG, center (of the corresponding circle) D along AG, and BD perpendicular to AG (see figure below). Then form the triangle ABG and the (segment of the) parabola ABG, as pictured. Al-Kuhi considered both these plane figures and the solids obtained by rotating them about the line BD (sweeping out a hemisphere, a cone, and a paraboloid). He announced in a letter to al-Sabi that the centers of gravity of the various planar and solid figures lie along BD at points P dividing BD in the following ratios:
(a) Triangle: DP/DB = 1/3.
(b) Cone: DP/DB = 1/4.
(c) Parabola: DP/DB = 2/5.
(d) Paraboloid: DP/DB = 2/6.
(e) Semicircle: DP/DB = 3/7.
(f) Hemisphere: DP/DB = 3/8.
Such calculations are great feats prior to the introduction of calculus, but they are not all correct. Use calculus to derive the correct results for the planar figures (triangle, parabola, semicircle). Here is an outline of one way to proceed: we may assume the circle has radius 1, and that the points A and D are (0, 0) and (1, 0). For any of the figures considered, let f(x) denote the height at x, and let A denote the total area, so that A = ??² f(x)dx. Calculus tells us that the point P is then (1, p) where
p = [??² (f(x)²/2) dx] / A
Use this formula to compute p for each of the three shapes. For the semicircle, you will find that your answer does not agree exactly with that of al-Kuhi: what value of ? would be implied by the semicircle answer? (Our reading from the Sourcebook, 572-573, is in fact al-Kuhi’s response to a contemporary who questioned the result of his calculation, since it seemed at variance with Archimedes’ approximations to ?. See 568-572 of the Sourcebook for more of this exchange.)
(2) This week we will study Ibn al-Haytham’s work on the volume of a paraboloid. Consider a parabola and a line perpendicular to its axis that intersects the parabola in points P and Q and thereby determines a parabolic segment. Consider the solid obtained by rotating the parabolic segment about the line. Also consider the cylinder obtained by rotating the rectangle whose top is PQ and whose bottom is obtained by translating PQ so that it is tangent to the parabola at the vertex. (See picture.) Show that the volume of the paraboloid is 8/15 of the volume of the cylinder in the following steps:
(a) Write down an equation of the parabola and an equation of the line through P and Q.
(b) Compute the volume of the cylinder (in terms of the parameters from the previous part).
(c) Set up and compute an integral that computes the volume of the paraboloid, and compare it to your cylinder calculation.

(1) Al-Kuhi studied the centers of gravity of the following series of shapes. Fix a semicircle ABG with diameter AG, center (of the corresponding circle) D along AG, and BD perpendicular to AG (see figure below). Then form the triangle ABG and the (segment of the) parabola ABG, as pictured. Al-Kuhi considered both these plane figures and the solids obtained by rotating them about the line BD (sweeping out a hemisphere, a cone, and a paraboloid). He announced in a letter to al-Sabi that the centers of gravity of the various planar and solid figures lie along BD at points P dividing BD in the following ratios:
(a) Triangle: DP/DB = 1/3.
(b) Cone: DP/DB = 1/4.
(c) Parabola: DP/DB = 2/5.
(d) Paraboloid: DP/DB = 2/6.
(e) Semicircle: DP/DB = 3/7.
(f) Hemisphere: DP/DB = 3/8.
Such calculations are great feats prior to the introduction of calculus, but they are not all correct. Use calculus to derive the correct results for the planar figures (triangle, parabola, semicircle). Here is an outline of one way to proceed: we may assume the circle has radius 1, and that the points A and D are (0, 0) and (1, 0). For any of the figures considered, let f(x) denote the height at x, and let A denote the total area, so that A = ??² f(x)dx. Calculus tells us that the point P is then (1, p) where
p = [??² (f(x)²/2) dx] / A
Use this formula to compute p for each of the three shapes. For the semicircle, you will find that your answer does not agree exactly with that of al-Kuhi: what value of ? would be implied by the semicircle answer? (Our reading from the Sourcebook, 572-573, is in fact al-Kuhi’s response to a contemporary who questioned the result of his calculation, since it seemed at variance with Archimedes’ approximations to ?. See 568-572 of the Sourcebook for more of this exchange.)
(2) This week we will study Ibn al-Haytham’s work on the volume of a paraboloid. Consider a parabola and a line perpendicular to its axis that intersects the parabola in points P and Q and thereby determines a parabolic segment. Consider the solid obtained by rotating the parabolic segment about the line. Also consider the cylinder obtained by rotating the rectangle whose top is PQ and whose bottom is obtained by translating PQ so that it is tangent to the parabola at the vertex. (See picture.) Show that the volume of the paraboloid is 8/15 of the volume of the cylinder in the following steps:
(a) Write down an equation of the parabola and an equation of the line through P and Q.
(b) Compute the volume of the cylinder (in terms of the parameters from the previous part).
(c) Set up and compute an integral that computes the volume of the paraboloid, and compare it to your cylinder calculation.

Added by Levi C.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Sri K

10/18/2022

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(1) Al-Kuhi studied the centers of gravity of the following series of shapes. Fix a semicircle ABG with diameter AG, center (of the corresponding circle) D along AG, and BD perpendicular to AG (see figure below). Then form the triangle ABG and the (segment of the) parabola ABG, as pictured. Al-Kuhi considered both these plane figures and the solids obtained by rotating them about the line BD (sweeping out a hemisphere, a cone, and a paraboloid). He announced in a letter to al-Sabi that the centers of gravity of the various planar and solid figures lie along BD at points P dividing BD in the following ratios: (a) Triangle: DP/DB = 1/3. (b) Cone: DP/DB = 1/4. (c) Parabola: DP/DB = 2/5. (d) Paraboloid: DP/DB = 2/6. (e) Semicircle: DP/DB = 3/7. (f) Hemisphere: DP/DB = 3/8. Such calculations are great feats prior to the introduction of calculus, but they are not all correct. Use calculus to derive the correct results for the planar figures (triangle, parabola, semicircle). Here is an outline of one way to proceed: we may assume the circle has radius 1, and that the points A and D are (0, 0) and (1, 0). For any of the figures considered, let f(x) denote the height at x, and let A denote the total area, so that A = ∫ f(x)dx. Calculus tells us that the point P is then (1, p) where p = [∫ (f(x)^2)/2 dx] / A Use this formula to compute p for each of the three shapes. For the semicircle, you will find that your answer does not agree exactly with that of al-Kuhi: what value of ̀π would be implied by the semicircle answer? (Our reading from the Sourcebook, 572-573, is in fact al-Kuhi’s response to a contemporary who questioned the result of his calculation, since it seemed at variance with Archimedes’ approximations to π. See 568-572 of the Sourcebook for more of this exchange.) (2) This week we will study Ibn al-Haytham’s work on the volume of a paraboloid. Consider a parabola and a line perpendicular to its axis that intersects the parabola in points P and Q and thereby determines a parabolic segment. Consider the solid obtained by rotating the parabolic segment about the line. Also consider the cylinder obtained by rotating the rectangle whose top is PQ and whose bottom is obtained by translating PQ so that it is tangent to the parabola at the vertex. (See picture.) Show that the volume of the paraboloid is 8/15 of the volume of the cylinder in the following steps: (a) Write down an equation of the parabola and an equation of the line through P and Q. (b) Compute the volume of the cylinder (in terms of the parameters from the previous part). (c) Set up and compute an integral that computes the volume of the paraboloid, and compare it to your cylinder calculation.