Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 8 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.
$\int _ { 3 } ^ { 4 } \sqrt [ 4 ] { x } d x$
A) The Trapezoidal rule gives $1.366878$ and Simpson's rule gives $1.435222$.
B) The Trapezoidal rule gives $- 1.366878$ and Simpson's rule gives $- 1.366906$.
C) The Trapezoidal rule gives $1.366878$ and Simpson's rule gives $1.366906$.
D) The Trapezoidal rule gives $1.366795$ and Simpson's rule gives $1.435222$.
E) The Trapezoidal rule gives $1.366795$ and Simpson's rule gives 1.366906.

Question

Quizplus · Accepted Answer

Using the Trapezoidal Rule with 8 subintervals, we have
\begin{align*}
\int _ { 3 } ^ { 4 } \sqrt [ 4 ] { x } d x &\approx \frac{1}{2}\left(\frac{1}{8}ight)\left[\left(3ight)^{1/4} + 2\left(3+\frac{1}{8}ight)^{1/4}+ 2\left(3+\frac{2}{8}ight)^{1/4}+ 2\left(3+\frac{3}{8}ight)^{1/4}+ 2\left(3+\frac{4}{8}ight)^{1/4}+ 2\left(3+\frac{5}{8}ight)^{1/4}+ 2\left(3+\frac{6}{8}ight)^{1/4} + \left(4ight)^{1/4}ight] \
& = 1.366878.
\end{align*}
Using Simpson's Rule with 8 subintervals, we have
\begin{align*}
\int _ { 3 } ^ { 4 } \sqrt [ 4 ] { x } d x &\approx \frac{1}{3}\left(\frac{1}{8}ight)\left[\left(3ight)^{1/4} + 3\left(3+\frac{1}{8}ight)^{1/4}+ 3\left(3+\frac{3}{8}ight)^{1/4}+ \left(4ight)^{1/4}ight]\
& = 1.366906.
\end{align*}
Using the exact value, we have
\begin{align*}
\int _ { 3 } ^ { 4 } \sqrt [ 4 ] { x } d x & = \frac{4}{5}\left(4^{5/4}-3^{5/4}ight) \
& = 1.366973.
\end{align*}
Comparing the approximations with the exact value, we see that the Trapezoidal Rule gives an error of about $0.000095$ and Simpson's Rule gives an error of about $0.000067$. The answer that is closest to the actual value and matching these errors is choice C.

Apply the Trapezoidal Rule and Simpson's Rule to Approximate the Value