Question

Find the error that occurs when the area between the curve of \(\ \sf y = x^3+ 1 \)

As well as, the x axis over the interval of: \(\ [0,1]\); using approximately the trapezoid rule with n equal to 4

Answer 1

Exact Area = \[\int _0^1 (x^3+1)dx\]

Answer 2

For trapezoid rule, split [0,1] into 4 equal parts:
\[
\Delta x = 0.25; ~~ x_0 = 0, ~~x_1 = 0.25, ~~ x_2 = 0.5, ~~ x_3 = 0.75 ~~ x_4 = 1.0\\
\text{Area = }\frac{\Delta x}{2}\{f(x_0+2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \} = ?
\]

Answer 3

I got 0.016

Answer 4

\[
\text{Exact Area = }\int _0^1 (x^3+1)dx =\left[x^4/4 + x\right]_0^1 = 1.25 \\
\text{Area using Trapezoidal Rule: } \\
\Delta x = 0.25; ~~ x_0 = 0, ~~x_1 = 0.25, ~~ x_2 = 0.5, ~~ x_3 = 0.75 ~~ x_4 = 1.0\\
\text{Area }\approx \frac{\Delta x}{2}\{f(x_0)+2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4) \} = \\
\frac{0.25}{2}\{1 + 2.03125 + 2.25 + 2.84375 + 2 \} = 1.265625
\]

Answer 5

Damn ;-;