Using trapezoidal rule, estimate the area under the curve

Question

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

Answer 1

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.