27a) Using trapezoidal rule with 6 strips, estimate the area under the curve y = 1/2 x2 - 2 between x = 2 and x = 8 and axis. (4marks) b) Use integration to evaluate the exact area under the curve (3marks) Find the percentage error in calculating the area using trapezoidal rule (3marks)

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Asked on August 11, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-08-11T12:07:48.132Z

Huu hapa ni utatuzi wa swali lako: a) Kutumia kanuni ya trapezoidal kukadiria eneo: Kazi ni y = (1)/(2)x^2 - 2. Vipimo ni x=2 hadi x=8. Idadi ya vipande (strips) ni n=6. Step 1: Tafuta upana wa kila kipande, h. h = (b-a)/(n) = (8-2)/(6) = (6)/(6) = 1 Step 2: Bainisha thamani za x kwa kila kipande na hesabu thamani za y zinazolingana. • x_0 = 2 y_0 = (1)/(2)(2^2) - 2 = 0 • x_1 = 3 y_1 = (1)/(2)(3^2) - 2 = 2.5 • x_2 = 4 y_2 = (1)/(2)(4^2) - 2 = 6 • x_3 = 5 y_3 = (1)/(2)(5^2) - 2 = 10.5 • x_4 = 6 y_4 = (1)/(2)(6^2) - 2 = 16 • x_5 = 7 y_5 = (1)/(2)(7^2) - 2 = 22.5 • x_6 = 8 y_6 = (1)/(2)(8^2) - 2 = 30 Step 3: Tumia kanuni ya trapezoidal. Eneo ≈ (h)/(2) [y_0 + y_n + 2(y_1 + y_2 + ... + y_n-1)] Eneo ≈ (1)/(2) [0 + 30 + 2(2.5 + 6 + 10.5 + 16 + 22.5)] Eneo ≈ (1)/(2) [30 + 2(57.5)] Eneo ≈ (1)/(2) [30 + 115] Eneo ≈ (1)/(2) [145] Step 4: Hesabu eneo lililokadiriwa. Eneo ≈ 72.5 Eneo lililokadiriwa ni 72.5 vitengo vya eneo. b) Kutumia integration kutafuta eneo halisi: Step 1: Unganisha kazi y = (1)/(2)x^2 - 2 kutoka x=2 hadi x=8. Eneo = _2^8 ((1)/(2)x^2 - 2)\,dx Eneo = [(1)/(2) · (x^3)/(3) - 2x]_2^8 Eneo = [(x^3)/(6) - 2x]_2^8 Step 2: Tathmini mipaka ya integration. Eneo = ((8^3)/(6) - 2(8)) - ((2^3)/(6) - 2(2)) Eneo = ((512)/(6) - 16) - ((8)/(6) - 4) Eneo = ((256)/(3) - (48)/(3)) - ((4)/(3) - (12)/(3)) Eneo = ((208)/(3)) - (-(8)/(3)) Eneo = (208)/(3) + (8)/(3) = (216)/(3) Eneo = 72 Eneo halisi ni 72 vitengo vya eneo. c) Tafuta asilimia ya makosa: Asilimia ya makosa huhesabiwa kwa kutumia formula: Asilimia ya Makosa = |Thamani Iliyokadiriwa - Thamani HalisiThamani Halisi| × 100\% Step 1: Badilisha thamani zilizopatikana. Asilimia ya Makosa = |(72.5 - 72)/(72)| × 100\% Asilimia ya Makosa = |(0.5)/(72)| × 100\% Asilimia ya Makosa = (0.5)/(72) × 100\% Asilimia ya Makosa = (50)/(72)\% Asilimia ya Makosa ≈ 0.6944\% Asilimia ya makosa ni 0.6944\%. Tuma swali linalofuata 📸

Accepted Answer · 2026-08-11T12:07:48.132Z

Huu hapa ni utatuzi wa swali lako: a) Kutumia kanuni ya trapezoidal kukadiria eneo: Kazi ni y = (1)/(2)x^2 - 2. Vipimo ni x=2 hadi x=8. Idadi ya vipande (strips) ni n=6. Step 1: Tafuta upana wa kila kipande, h. h = (b-a)/(n) = (8-2)/(6) = (6)/(6) = 1 Step 2: Bainisha thamani za x kwa kila kipande na hesabu thamani za y zinazolingana. • x_0 = 2 y_0 = (1)/(2)(2^2) - 2 = 0 • x_1 = 3 y_1 = (1)/(2)(3^2) - 2 = 2.5 • x_2 = 4 y_2 = (1)/(2)(4^2) - 2 = 6 • x_3 = 5 y_3 = (1)/(2)(5^2) - 2 = 10.5 • x_4 = 6 y_4 = (1)/(2)(6^2) - 2 = 16 • x_5 = 7 y_5 = (1)/(2)(7^2) - 2 = 22.5 • x_6 = 8 y_6 = (1)/(2)(8^2) - 2 = 30 Step 3: Tumia kanuni ya trapezoidal. Eneo ≈ (h)/(2) [y_0 + y_n + 2(y_1 + y_2 + ... + y_n-1)] Eneo ≈ (1)/(2) [0 + 30 + 2(2.5 + 6 + 10.5 + 16 + 22.5)] Eneo ≈ (1)/(2) [30 + 2(57.5)] Eneo ≈ (1)/(2) [30 + 115] Eneo ≈ (1)/(2) [145] Step 4: Hesabu eneo lililokadiriwa. Eneo ≈ 72.5 Eneo lililokadiriwa ni 72.5 vitengo vya eneo. b) Kutumia integration kutafuta eneo halisi: Step 1: Unganisha kazi y = (1)/(2)x^2 - 2 kutoka x=2 hadi x=8. Eneo = _2^8 ((1)/(2)x^2 - 2)\,dx Eneo = [(1)/(2) · (x^3)/(3) - 2x]_2^8 Eneo = [(x^3)/(6) - 2x]_2^8 Step 2: Tathmini mipaka ya integration. Eneo = ((8^3)/(6) - 2(8)) - ((2^3)/(6) - 2(2)) Eneo = ((512)/(6) - 16) - ((8)/(6) - 4) Eneo = ((256)/(3) - (48)/(3)) - ((4)/(3) - (12)/(3)) Eneo = ((208)/(3)) - (-(8)/(3)) Eneo = (208)/(3) + (8)/(3) = (216)/(3) Eneo = 72 Eneo halisi ni 72 vitengo vya eneo. c) Tafuta asilimia ya makosa: Asilimia ya makosa huhesabiwa kwa kutumia formula: Asilimia ya Makosa = |Thamani Iliyokadiriwa - Thamani HalisiThamani Halisi| × 100\% Step 1: Badilisha thamani zilizopatikana. Asilimia ya Makosa = |(72.5 - 72)/(72)| × 100\% Asilimia ya Makosa = |(0.5)/(72)| × 100\% Asilimia ya Makosa = (0.5)/(72) × 100\% Asilimia ya Makosa = (50)/(72)\% Asilimia ya Makosa ≈ 0.6944\% Asilimia ya makosa ni 0.6944\%. Tuma swali linalofuata 📸

27a) Using trapezoidal rule with 6 strips, estimate the area under the curve y = 1/2 x2 - 2 between x = 2 and x = 8 and axis. (4marks) b) Use integration to evaluate the exact area under the curve (3marks) Find the percentage error in calculating the area using trapezoidal rule (3marks)

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27a) Using trapezoidal rule with 6 strips, estimate the area under the curve y = 1/2 x2 - 2 between x = 2 and x = 8 and axis. (4marks) b) Use integration to evaluate the exact area under the curve (3marks) Find the percentage error in calculating the area using trapezoidal rule (3marks)

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