Use the Midpoint Rule with n = 2 to approximate the integral $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$
A) $\frac { 1 } { 4 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 7 } { 32 }$
D) $\frac { 3 } { 2 }$
E) $\frac { 5 } { 16 }$
F) $\frac { 5 } { 32 }$
G) $\frac { 7 } { 16 }$
H) $\frac { 5 } { 2 }$

Question

Use the Midpoint Rule with n = 2 to approximate the integral $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$ 
A) $\frac { 1 } { 4 }$ 
B) $\frac { 1 } { 2 }$ 
C) $\frac { 7 } { 32 }$ 
D) $\frac { 3 } { 2 }$ 
E) $\frac { 5 } { 16 }$ 
F) $\frac { 5 } { 32 }$ 
G) $\frac { 7 } { 16 }$ 
H) $\frac { 5 } { 2 }$

Quizplus · Accepted Answer

The Midpoint Rule for approximating integrals is given by $M_n = \Delta x \sum_{i=1}^{n} f(\bar{x}_i)$, where $\Delta x = \frac{b-a}{n}$ and $\bar{x}_i = a + \left(i-\frac{1}{2}\right)\Delta x$. For $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$ with $n=2$, $\Delta x = \frac{1-0}{2} = 0.5$. The midpoints are $0.25$ and $0.75$. Thus, $M_2 = 0.5[f(0.25) + f(0.75)] = 0.5[(0.25)^3 + (0.75)^3] = 0.5[0.015625 + 0.421875] = 0.5 \times 0.4375 = 0.21875$, which is closest to $\frac{7}{32}$ when converted to a fraction.

Use the Midpoint Rule with N = 2 to Approximate ∫01x3dx\int _ { 0 } ^ { 1 } x ^ { 3 } d x∫01​x3dx

Use the Midpoint Rule with N = 2 to Approximate $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$