Section 5. 10. Find the areas and circumferences of the diameters are 27 feet, 48 feet, 35 feet, 17 circles, whose feet, and 265 feet. Section 6. 11. Find the areas and circumferences of the circles, whose diameters are 25 feet, 45 feet, 49 feet, 27 feet 5 inches, and 37 feet 7 inches. Section 7. 12. The diameter of a piston for a steam- engine cylinder is 4 feet 8 inches, and the mean pressure of steam upon it is 10 lbs. per square inch; required the pressure, in tons, on the piston. 13. The mean pressure of steam in a cylinder is 16 lbs. per inch, and the total pressure was observed to be 6 tons; required the diameter of the piston. 14. The area of a circle is 20.129 square feet; find the circumference and diameter. Section 8. 15. Find the area of the ellipses, whose diameters are 16 feet, 4 feet; 26 feet, 9 feet; 25 feet, 14 feet; 32 feet, and 22 feet. Section 9. 16. Find the area of an elliptical field, whose axes are 27 and 3 chains. 17. The area of an elliptical garden is 84 square yards, and one of its axes is 18 yards; required the other. 18. What is the diameter of a circular field, and also the side of a square field, each of which contains an acre ? Section 10. 19. Find the area of a parabola, whose abscissa is 8 feet, and the double ordinate 15 feet. 20. Find the area of the parabolas, whose abscissa and double ordinates are as follow: 213 feet, 32 feet; 25 feet, 24} feet; 25 feet, and 41 feet. Section 11. 21. The equatorial and polar diameters of the earth are 7926 miles, and 7899 miles; find the difference between a section passing through the equator, and another passing through a meridian. 22. Find the area of a parabola, whose abscissa and double ordinate are 524 feet, and 36 feet. Section 12. (Requires algebra.) 23. The area of a circle is equal to 6 times its circumference ; find its area and diameter. 24. The area of a circle is equal to the circumference and radius; find its area. 25. If 4 be added to the circumference of a circle, the sum will be the area; find its area. 26. The area of a circle is equal to 5 times its radius added to the circumference of a circle whose radius is 2; find the area. 27. The area of a circle is equal to its circumference, and area of an equilateral triangle inscribed within it; find the area of the circle and triangle. PROBLEM 5. To find the area of any curve by Simpson's Rule. Divide A Y into an even number of equal parts in the points B, C, &c., then draw the ordinates A A', BB', C C', &c. perpendicular to A Y, and meeting the curve in the points A', B'C'' &c ABC & Y B', C', &c. The number of equal parts into which A Y is divided must depend entirely upon the degree of curvature of the curve, and the accuracy with which the area is required. Generally, the greater the number of parts into which A Y is divided the greater will be the accuracy of the area. Rule.-Take half the sum of the first and last ordinates for the first sum. Take twice the sum of all the even ordinates, for the second sum. Take the sum of all the uneven ordinates, omitting the first and last ordinates, for a third sum. Add the first, second, and third sums together, and multiply this sum by the common interval A B, and the product will be the area of the curve. FORMULA. Put A A'a,, B B' = α2, C C' = α 3, &c. + 2 (a1⁄2 +а + a6 + &c.) + (a3 + az + α, + §c.) 4 1. The ordinates of a curve are 1, 2, 4, 6, 8, 7, 5 feet, and the common interval 2 feet; find the area. 2. The ordinates of a curve are 2, 3, 5, 8, 10, 14, 13, 9, 6, 3, 1 feet, and the common interval 9 inches; find the area. Section 2. 3. The ordinates of a curve are 0, 11, 31, 4, 6, 7, 8, 10, 7, 5, 3, 2, 1 feet respectively, and the common interval 1 foot 3 inches; find its area. 4. The ordinates of a curve are 2, 8, 11, 14, 16, 13, 14, 11, 8, 7, 5, 3, 0 feet respectively, and the common interval is 1 foot 9 inches; find its area. Section 3. 5. The ordinates of a curve are 0, 2, 31, 51, 6, 7, 8, 9, 74, 63, 5, 3, 2, 1, 0 feet respectively; find the area when the common interval is 1 foot 3 inches. 6. The ordinates of a curve are 5, 21.9, 26.9, 28.15, 28.9, 29.4, 29.6, 29.7, 29.7, 29.7, 29.7, 29.7, 29.6, 29.5, 29.3, 28.95, 28.5, 27.75, 26.9, 25.9, 24.5 feet respectively; find the area when the common interval is 10 feet. Section 4. 7. The ordinates of a curve are 3, 4, 5, 7, 6, 2, 1 feet, and common interval 2 feet; required the area. 8. The ordinates of a curve are 1, 2. 4, 7, 9, 6, 5, 3, 2 feet, and common interval 1 foot 6 inches; find its area. 9. The ordinates of a curve are 17.50, 380.74, 672.8, 790.06, 855.21, 894.53, 909.07, 919.72, 921.26, 922.8, 922.8, 916.65, 907.53, 892.25, 874.37, 850.13, 815.83, 761.6, 690.52, 569.95, 185.84 feet respectively; find the area when the common interval is 10 feet. Section 5. 10. The ordinates of a curve are 3, 5, 9, 4, 2 feet; and common interval 3 feet; find its area. 11. The ordinates of a curve are 5, 9, 12, 14, 20, 20, 20, 18, 16, 11, 8 feet, and common interval 6 feet; find the area. 12. The ordinates of a curve are 6, 14.65, 19.4, 22.55, 24.6, 25.95, 26.8. 27.1, 27.3, 27.3, 27.3, 27.25, 27.2, 26.95, 26.5, 25.75, 24.6, 23.05, 21.1, 18.5, 15.4, 11.1, 3.9 feet, and the common interval is 9 feet; find its area. Section 6. 13. The ordinate of a curve are 5, 21, 26, 28, 28, 29, 27, 26, 25 feet, and the common interval 10 feet; required the area. 4. The ordinates of a curve are 9, 10, 12, 12, 12, 11, 8 feet, and the common interval 3 feet; find the area. 15. The ordinates of a curve are 43.5, 266.28, 453.13, 587.34, 659.93, 709.65, 741.82, 754.06, 760.86, 760.86, 760.86, 759.51, 758.16, 748.63, 732.11, 708.08, 668.94, 618.16, 549.93, 456.09, 3.39.27, 187.72, 18.5 feet, the commo interval is 9 feet; required the area. Section 7. 16. The ordinates of a curve are 3, 9, 10, 12, 14, 15, 15, 15, 13, 11, 9, 4, 0 feet; required the area when the common interval is 2 feet. 17. The ordinates of a curve are 604.96, 7531.28, 13637.29, 17577.27, 19165.04, 20323.38, 20972.6, 21280.59, 21359.7, 21359.7, 21359.7, 21359.7, 21359.7, 21093.23, 20673.81, 20295.02, 19482.6, 18619.13, 17241.16, 15643.06, 13137.14, 9045.65, 902.5 feet; find the area when the common interval is 9 feet. Section 8. 18. The ordinates of a curve are 10.85, 21.65, 25, 26.4, 26.95, 27.25, 27.4, 27.5, 27.5, 27.5, 27.5, 27.5, 27.5, 27.35, 27.2, 27.1, 26.8, 26.5, 26.05, 25.5, 24.7, 23.15, 18.5 feet; find the area when the common interval is 9 feet. 19. The ordinates of a curve are 0, 4, 6, 7, 7, 7. 5, 3, 0 feet; find the area when the common interval is 2 feet 6 inches. Section 9. 20. The ordinates of a curve are 3.2, 13, 18.4, 21.75, 23.9, 25.15, 26, 26.5, 26.8, 26.9, 26.9, 26.7, 26.3, 25.7, 24.9, 23.65, 22, 19.6, 16.5, 11.9, 3.5 feet; required the area when the common interval is 10 feet. 21. The ordinates of a curve are 0, 2, 4, 7, 8, 12, 14, 17, 16, 13, 11, 9, 7, 6, 4 feet, and the common interval 6 feet; required the area. 22. The ordinates of a curve are 0, 1, 3, 4, 5, 6, 7, 8, 9, 4, 2, 1, 0 feet, and the common interval 8 feet; find its area. PROBLEM 6. To find the centre of gravity of any curve by Simpson's Rule. |