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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)

Mathematics

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Graph of y = 1/2x² − 2

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Key points of y = 1/2x² − 2
Pointxy
x-intercept20

Step-by-step solution

7 steps

Huu hapa ni utatuzi wa swali lako:

Part a· 4 steps

Kutumia kanuni ya trapezoidal kukadiria eneo

Kazi ni y=12x2−2y = \frac{1}{2}x^2 - 2. Vipimo ni x=2x=2 hadi x=8x=8. Idadi ya vipande (strips) ni n=6n=6.

  1. Tafuta upana wa kila kipande, hh.

    h=b−an=8−26=66=1h = \frac{b-a}{n} = \frac{8-2}{6} = \frac{6}{6} = 1

  2. Bainisha thamani za xx kwa kila kipande na hesabu thamani za yy zinazolingana.

    • x0=2  ⟹  y0=12(22)−2=0x_0 = 2 \implies y_0 = \frac{1}{2}(2^2) - 2 = 0
    • x1=3  ⟹  y1=12(32)−2=2.5x_1 = 3 \implies y_1 = \frac{1}{2}(3^2) - 2 = 2.5
    • x2=4  ⟹  y2=12(42)−2=6x_2 = 4 \implies y_2 = \frac{1}{2}(4^2) - 2 = 6
    • x3=5  ⟹  y3=12(52)−2=10.5x_3 = 5 \implies y_3 = \frac{1}{2}(5^2) - 2 = 10.5
    • x4=6  ⟹  y4=12(62)−2=16x_4 = 6 \implies y_4 = \frac{1}{2}(6^2) - 2 = 16
    • x5=7  ⟹  y5=12(72)−2=22.5x_5 = 7 \implies y_5 = \frac{1}{2}(7^2) - 2 = 22.5
    • x6=8  ⟹  y6=12(82)−2=30x_6 = 8 \implies y_6 = \frac{1}{2}(8^2) - 2 = 30

  3. Tumia kanuni ya trapezoidal.

    Eneo ≈h2[y0+yn+2(y1+y2+...+yn−1)]\approx \frac{h}{2} [y_0 + y_n + 2(y_1 + y_2 + ... + y_{n-1})]

    Eneo≈12[0+30+2(2.5+6+10.5+16+22.5)]\text{Eneo} \approx \frac{1}{2} [0 + 30 + 2(2.5 + 6 + 10.5 + 16 + 22.5)] Eneo≈12[30+2(57.5)]\text{Eneo} \approx \frac{1}{2} [30 + 2(57.5)] Eneo≈12[30+115]\text{Eneo} \approx \frac{1}{2} [30 + 115] Eneo≈12[145]\text{Eneo} \approx \frac{1}{2} [145]

  4. Hesabu eneo lililokadiriwa.

    Eneo≈72.5\text{Eneo} \approx 72.5

    Eneo lililokadiriwa ni 72.5 vitengo vya eneo\boxed{\text{72.5 vitengo vya eneo}}.

Part b· 2 steps

Kutumia integration kutafuta eneo halisi

  1. Unganisha kazi y=12x2−2y = \frac{1}{2}x^2 - 2 kutoka x=2x=2 hadi x=8x=8.

    Eneo=∫28(12x2−2) dx\text{Eneo} = \int_{2}^{8} \left(\frac{1}{2}x^2 - 2\right)\,dx Eneo=[12⋅x33−2x]28\text{Eneo} = \left[\frac{1}{2} \cdot \frac{x^3}{3} - 2x\right]_{2}^{8} Eneo=[x36−2x]28\text{Eneo} = \left[\frac{x^3}{6} - 2x\right]_{2}^{8}

  2. Tathmini mipaka ya integration.

    Eneo=(836−2(8))−(236−2(2))\text{Eneo} = \left(\frac{8^3}{6} - 2(8)\right) - \left(\frac{2^3}{6} - 2(2)\right) Eneo=(5126−16)−(86−4)\text{Eneo} = \left(\frac{512}{6} - 16\right) - \left(\frac{8}{6} - 4\right) Eneo=(2563−483)−(43−123)\text{Eneo} = \left(\frac{256}{3} - \frac{48}{3}\right) - \left(\frac{4}{3} - \frac{12}{3}\right) Eneo=(2083)−(−83)\text{Eneo} = \left(\frac{208}{3}\right) - \left(-\frac{8}{3}\right) Eneo=2083+83=2163\text{Eneo} = \frac{208}{3} + \frac{8}{3} = \frac{216}{3} Eneo=72\text{Eneo} = 72

    Eneo halisi ni 72 vitengo vya eneo\boxed{\text{72 vitengo vya eneo}}.

Part c· 1 step

Tafuta asilimia ya makosa

Asilimia ya makosa huhesabiwa kwa kutumia formula:

Asilimia ya Makosa=∣Thamani Iliyokadiriwa−Thamani HalisiThamani Halisi∣×100%\text{Asilimia ya Makosa} = \left|\frac{\text{Thamani Iliyokadiriwa} - \text{Thamani Halisi}}{\text{Thamani Halisi}}\right| \times 100\%

  1. Badilisha thamani zilizopatikana.

    Asilimia ya Makosa=∣72.5−7272∣×100%\text{Asilimia ya Makosa} = \left|\frac{72.5 - 72}{72}\right| \times 100\% Asilimia ya Makosa=∣0.572∣×100%\text{Asilimia ya Makosa} = \left|\frac{0.5}{72}\right| \times 100\% Asilimia ya Makosa=0.572×100%\text{Asilimia ya Makosa} = \frac{0.5}{72} \times 100\% Asilimia ya Makosa=5072%\text{Asilimia ya Makosa} = \frac{50}{72}\% Asilimia ya Makosa≈0.6944%\text{Asilimia ya Makosa} \approx 0.6944\%

    Asilimia ya makosa ni 0.6944%\boxed{\text{0.6944\%}}.

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