Using trapezoidal rule, estimate the area under the curve
Y=1/2x^2-2 between x=2 and x=8 and x-axis. Use 6 strips
(a) use integration to evaluate the exact area under the curve
Using trapezoidal rule with 6 strips, we have:
width of each strip = (8-2)/6 = 1.0
x-values of the strip endpoints:
strip 1: 2.0, 3.0
strip 2: 3.0, 4.0
strip 3: 4.0, 5.0
strip 4: 5.0, 6.0
strip 5: 6.0, 7.0
strip 6: 7.0, 8.0
Using the formula for trapezoidal rule:
Area ≈ (width/2) [(f(x1) + f(x2)) + (f(x2) + f(x3)) + ... + (f(x5) + f(x6))]
where f(x) = 1/2x^2 - 2 is the function we are integrating.
Plugging in the x-values and evaluating the function at those points, we get:
Area ≈ (1/2) [(f(2.0) + f(3.0)) + (f(3.0) + f(4.0)) + (f(4.0) + f(5.0)) + (f(5.0) + f(6.0)) + (f(6.0) + f(7.0)) + (f(7.0) + f(8.0))]
Area ≈ (1/2) [(0 + (-1/2)) + (-1/2 + 2) + (2 + 9/2) + (9/2 + 16/2) + (16/2 + 27/2) + (27/2 + 30)]
Area ≈ 63
Using integration to find the exact area under the curve:
∫ 1/2x^2 - 2 dx = 1/6x^3 - 2x + C
Evaluating the definite integral from x=2 to x=8:
Area = 1/6(8^3) - 2(8) - [1/6(2^3) - 2(2)]
Area = 64/3 - 4/3
Area = 60/3 = 20
Therefore, the exact area under the curve is 20, and the estimated area using trapezoidal rule with 6 strips is roughly 63.