Question 4.7.1: Using the Midpoint Rule Write out the Midpoint Rule approxim...

Using the Midpoint Rule
Write out the Midpoint Rule approximation of \int_0^1 3 x^2 d x with n = 4.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

For n = 4, the regular partition of the interval [0, 1] is x0 = 0, x_1=\frac{1}{4}, x_2=\frac{1}{2}, x_3=\frac{3}{4} and x4 = 1, The midpoints are then c_1=\frac{1}{8}, c_2=\frac{3}{8}, c_3=\frac{5}{8} and c_4=\frac{7}{8}.

\left[f\left(\frac{1}{8}\right)+f\left(\frac{3}{8}\right)+f\left(\frac{5}{8}\right)+f\left(\frac{7}{8}\right)\right]\left(\frac{1}{4}\right)=\left(\frac{3}{64}+\frac{27}{64}+\frac{75}{64}+\frac{147}{64}\right)\left(\frac{1}{4}\right)

= \frac{252}{256}=0.984375.

Related Answered Questions

Question: 4.5.2

Verified Answer:

Observe that since f(x)=x^{1 / 2}-x^{-2}[/l...
Question: 4.4.6

Verified Answer:

From (4.3), we have f_{ ave }=\lim _{n \rig...
Question: 4.8.3

Verified Answer:

We can verify this as follows. First, recall that ...