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.
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Here's how to solve parts (b), (c), and (d) of the problem. b) Using the trapezium rule, with 8 strips, estimate the area bounded by the curve y = 4 - (1)/(4)x^2 and the x-axis for -4 x 4. Step 1: Calculate the width of each strip (h). The interval is from x = -4 to x = 4, and the number of strips is n = 8. h = upper limit - lower limitnumber of strips = (4 - (-4))/(8) = (8)/(8) = 1 Step 2: Determine the x-values for the ordinates and calculate the corresponding y-values using y = 4 - (1)/(4)x^2. The x-values are x_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: y_0 = 4 - (1)/(4)(-4)^2 = 4 - (1)/(4)(16) = 4 - 4 = 0 y_1 = 4 - (1)/(4)(-3)^2 = 4 - (1)/(4)(9) = 4 - 2.25 = 1.75 y_2 = 4 - (1)/(4)(-2)^2 = 4 - (1)/(4)(4) = 4 - 1 = 3 y_3 = 4 - (1)/(4)(-1)^2 = 4 - (1)/(4)(1) = 4 - 0.25 = 3.75 y_4 = 4 - (1)/(4)(0)^2 = 4 - 0 = 4 y_5 = 4 - (1)/(4)(1)^2 = 4 - (1)/(4)(1) = 4 - 0.25 = 3.75 y_6 = 4 - (1)/(4)(2)^2 = 4 - (1)/(4)(4) = 4 - 1 = 3 y_7 = 4 - (1)/(4)(3)^2 = 4 - (1)/(4)(9) = 4 - 2.25 = 1.75 y_8 = 4 - (1)/(4)(4)^2 = 4 - (1)/(4)(16) = 4 - 4 = 0 Step 3: Apply the trapezium rule formula. Area ≈ (h)/(2) [y_0 + y_n + 2(y_1 + y_2 + + y_n-1)] Area ≈ (1)/(2) [y_0 + y_8 + 2(y_1 + y_2 + y_3 + y_4 + y_5 + y_6 + y_7)] Area ≈ (1)/(2) [0 + 0 + 2(1.75 + 3 + 3.75 + 4 + 3.75 + 3 + 1.75)] Area ≈ (1)/(2) [0 + 2(21)] Area (1)/(2) [42] Area ≈ 21 The estimated area using the trapezium rule is 21 square units. c) Find the area estimated in part (b) above by integration. Step 1: Set up the definite integral. The actual area is found by integrating the function y = 4 - (1)/(4)x^2 from x = -4 to x = 4. Actual Area = _-4^4 (4 - (1)/(4)x^2)\,dx Step 2: Integrate the function. (4 - (1)/(4)x^2)\,dx = 4x - (1)/(4) · (x^3)/(3) + C = 4x - (x^3)/(12) + C Step 3: Evaluate the definite integral using the limits. Actual Area = [4x - (x^3)/(12)]_-4^4 = (4(4) - (4^3)/(12)) - (4(-4) - ((-4)^3)/(12)) = (16 - (64)/(12)) - (-16 - (-64)/(12)) = (16 - (16)/(3)) - (-16 + (16)/(3)) = 16 - (16)/(3) + 16 - (16)/(3) = 32 - (32)/(3) = (96 - 32)/(3) = (64)/(3) The actual area by integration is (64)/(3) square units. d) Calculate the percentage error in estimating the area using the trapezium rule. Step 1: State the estimated and actual values. Estimated Area (from part b) = 21 Actual Area (from part c) = (64)/(3) Step 2: Calculate the absolute error. Absolute Error = |Estimated Area - Actual Area| = |21 - (64)/(3)| = |(63)/(3) - (64)/(3)| = |-(1)/(3)| = (1)/(3) Step 3: Calculate the percentage error. Percentage Error = Absolute ErrorActual Area × 100\% = (1)/(3)(64)/(3) × 100\% = (1)/(3) × (3)/(64) × 100\% = (1)/(64) × 100\% = 1.5625\% Rounding to two decimal places, the percentage error is 1.56\%. That's 2 down. 3 left today — send the next one.