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Using the trapezium rule, with 8 strips, estimate the area bounded by the curve y = 4 − 1/4x2 and the x-axis for -4 <= x <= 4.

Mathematics

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Graph of y = 4 − 1/4x²

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Key points of y = 4 − 1/4x²
Pointxy
x-intercept−40
Maximum04
x-intercept40

Step-by-step solution

9 steps

Here's how to solve parts (b), (c), and (d) of the problem.

Part b· 3 steps

Using the trapezium rule, with 8 strips, estimate the area bounded by the curve y=4−14x2y = 4 - \frac{1}{4}x^2 and the x-axis for −4≤x≤4-4 \le x \le 4.

  1. Calculate the width of each strip (hh).

    The interval is from x=−4x = -4 to x=4x = 4, and the number of strips is n=8n = 8.

    h=upper limit−lower limitnumber of strips=4−(−4)8=88=1h = \frac{\text{upper limit} - \text{lower limit}}{\text{number of strips}} = \frac{4 - (-4)}{8} = \frac{8}{8} = 1

  2. Determine the x-values for the ordinates and calculate the corresponding y-values using y=4−14x2y = 4 - \frac{1}{4}x^2.

    The x-values are x0=−4,x1=−3,x2=−2,x3=−1,x4=0,x5=1,x6=2,x7=3,x8=4x_0 = -4, x_1 = -3, x_2 = -2, x_3 = -1, x_4 = 0, x_5 = 1, x_6 = 2, x_7 = 3, x_8 = 4.
    The corresponding y-values are:

    y0=4−14(−4)2=4−14(16)=4−4=0y_0 = 4 - \frac{1}{4}(-4)^2 = 4 - \frac{1}{4}(16) = 4 - 4 = 0 y1=4−14(−3)2=4−14(9)=4−2.25=1.75y_1 = 4 - \frac{1}{4}(-3)^2 = 4 - \frac{1}{4}(9) = 4 - 2.25 = 1.75 y2=4−14(−2)2=4−14(4)=4−1=3y_2 = 4 - \frac{1}{4}(-2)^2 = 4 - \frac{1}{4}(4) = 4 - 1 = 3 y3=4−14(−1)2=4−14(1)=4−0.25=3.75y_3 = 4 - \frac{1}{4}(-1)^2 = 4 - \frac{1}{4}(1) = 4 - 0.25 = 3.75 y4=4−14(0)2=4−0=4y_4 = 4 - \frac{1}{4}(0)^2 = 4 - 0 = 4 y5=4−14(1)2=4−14(1)=4−0.25=3.75y_5 = 4 - \frac{1}{4}(1)^2 = 4 - \frac{1}{4}(1) = 4 - 0.25 = 3.75 y6=4−14(2)2=4−14(4)=4−1=3y_6 = 4 - \frac{1}{4}(2)^2 = 4 - \frac{1}{4}(4) = 4 - 1 = 3 y7=4−14(3)2=4−14(9)=4−2.25=1.75y_7 = 4 - \frac{1}{4}(3)^2 = 4 - \frac{1}{4}(9) = 4 - 2.25 = 1.75 y8=4−14(4)2=4−14(16)=4−4=0y_8 = 4 - \frac{1}{4}(4)^2 = 4 - \frac{1}{4}(16) = 4 - 4 = 0

  3. Apply the trapezium rule formula.

    Area≈h2[y0+yn+2(y1+y2+⋯+yn−1)]\text{Area} \approx \frac{h}{2} [y_0 + y_n + 2(y_1 + y_2 + \dots + y_{n-1})] Area≈12[y0+y8+2(y1+y2+y3+y4+y5+y6+y7)]\text{Area} \approx \frac{1}{2} [y_0 + y_8 + 2(y_1 + y_2 + y_3 + y_4 + y_5 + y_6 + y_7)] Area≈12[0+0+2(1.75+3+3.75+4+3.75+3+1.75)]\text{Area} \approx \frac{1}{2} [0 + 0 + 2(1.75 + 3 + 3.75 + 4 + 3.75 + 3 + 1.75)] Area≈12[0+2(21)]\text{Area} \approx \frac{1}{2} [0 + 2(21)] Area12[42]\text{Area} \frac{1}{2} [42] Area≈21\text{Area} \approx 21

    The estimated area using the trapezium rule is 21 square units\boxed{\text{21 square units}}.

Part c· 3 steps

Find the area estimated in part (b) above by integration.

  1. Set up the definite integral.

    The actual area is found by integrating the function y=4−14x2y = 4 - \frac{1}{4}x^2 from x=−4x = -4 to x=4x = 4.

    Actual Area=∫−44(4−14x2) dx\text{Actual Area} = \int_{-4}^{4} \left(4 - \frac{1}{4}x^2\right)\,dx

  2. Integrate the function.

    ∫(4−14x2) dx=4x−14⋅x33+C=4x−x312+C\int \left(4 - \frac{1}{4}x^2\right)\,dx = 4x - \frac{1}{4} \cdot \frac{x^3}{3} + C = 4x - \frac{x^3}{12} + C

  3. Evaluate the definite integral using the limits.

    Actual Area=[4x−x312]−44\text{Actual Area} = \left[4x - \frac{x^3}{12}\right]_{-4}^{4} =(4(4)−4312)−(4(−4)−(−4)312)= \left(4(4) - \frac{4^3}{12}\right) - \left(4(-4) - \frac{(-4)^3}{12}\right) =(16−6412)−(−16−−6412)= \left(16 - \frac{64}{12}\right) - \left(-16 - \frac{-64}{12}\right) =(16−163)−(−16+163)= \left(16 - \frac{16}{3}\right) - \left(-16 + \frac{16}{3}\right) =16−163+16−163= 16 - \frac{16}{3} + 16 - \frac{16}{3} =32−323= 32 - \frac{32}{3} =96−323= \frac{96 - 32}{3} =643= \frac{64}{3}

    The actual area by integration is 643 square units\boxed{\frac{64}{3} \text{ square units}}.

Part d· 3 steps

Calculate the percentage error in estimating the area using the trapezium rule.

  1. State the estimated and actual values.

    Estimated Area (from part b) = 21
    Actual Area (from part c) = 643\frac{64}{3}

  2. Calculate the absolute error.

    Absolute Error=∣Estimated Area−Actual Area∣\text{Absolute Error} = \left|\text{Estimated Area} - \text{Actual Area}\right| =∣21−643∣=∣633−643∣=∣−13∣=13= \left|21 - \frac{64}{3}\right| = \left|\frac{63}{3} - \frac{64}{3}\right| = \left|-\frac{1}{3}\right| = \frac{1}{3}

  3. Calculate the percentage error.

    Percentage Error=Absolute ErrorActual Area×100%\text{Percentage Error} = \frac{\text{Absolute Error}}{\text{Actual Area}} \times 100\% =13643×100%= \frac{\frac{1}{3}}{\frac{64}{3}} \times 100\% =13×364×100%= \frac{1}{3} \times \frac{3}{64} \times 100\% =164×100%= \frac{1}{64} \times 100\% =1.5625%= 1.5625\%

    Rounding to two decimal places, the percentage error is 1.56%\boxed{\text{1.56\%}}.

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