A rectangular swimming pool is 36 ft wide by 75 ft long. The table gives depths (d) from x = 0 at the shallow end to the diving end. Use the Trapezoidal Rule with n = 15 to estimate the volume of the pool.
V = ∫ [0,75] of 36*d(x)dx
(x) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
(d) 4 6.2 7.2 7.9 8.5 9 9.5 9.9 10.3 10.7 11.1 11.4 11.7 12.1 12.4 12.7
Ans: 26325
I already know the answer but how do they get this answer, please show the steps. Thanks.
V = ∫ [0,75] of 36*d(x)dx
(x) 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
(d) 4 6.2 7.2 7.9 8.5 9 9.5 9.9 10.3 10.7 11.1 11.4 11.7 12.1 12.4 12.7
Ans: 26325
I already know the answer but how do they get this answer, please show the steps. Thanks.
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∆x = 75/15 = 5
i . . . xᵢ . . . dᵢ
0 . . . 0 . . . 4
1 . . . 5 . . . 6.2
2 . . . 10 . . 7.2
3 . . . 15 . . 7.9
...
14 . . 70 . . 12.4
15 . . 75 . . 12.7
V ≈ ∑[i=1 to 15] 36 (dᵢ₋₁ + dᵢ)/2 ∆x
V ≈ 36 ∆x (1/2 d₀ + d₁ + d₂ + d₃ + . . . + d₁₄ + 1/2 d₁₅)
V ≈ 36 * 5 (1/2 (4) + 6.2 + 7.2 + 7.9 + 8.5 + 9 + 9.5 + 9.9 + 10.3 + 10.7 + 11.1 + 11.4 + 11.7 + 12.1 + 12.4 + 1/2 (12.7))
V ≈ 26325
i . . . xᵢ . . . dᵢ
0 . . . 0 . . . 4
1 . . . 5 . . . 6.2
2 . . . 10 . . 7.2
3 . . . 15 . . 7.9
...
14 . . 70 . . 12.4
15 . . 75 . . 12.7
V ≈ ∑[i=1 to 15] 36 (dᵢ₋₁ + dᵢ)/2 ∆x
V ≈ 36 ∆x (1/2 d₀ + d₁ + d₂ + d₃ + . . . + d₁₄ + 1/2 d₁₅)
V ≈ 36 * 5 (1/2 (4) + 6.2 + 7.2 + 7.9 + 8.5 + 9 + 9.5 + 9.9 + 10.3 + 10.7 + 11.1 + 11.4 + 11.7 + 12.1 + 12.4 + 1/2 (12.7))
V ≈ 26325