Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 20 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.
$\int _ { 1 } ^ { 2 } \left( 4 z ^ { 2 } + 7 \right) d z$
A) The Trapezoidal rule gives $16.3400$ and Simpson's rule gives 17,1518 .
B) The Trapezoidal rule gives $16.3400$ and Simpson's rule gives 16.3333.
C) The Trapezoidal rule gives $- 16.3350$ and Simpson's rule gives $- 16.3333$.
D) The Trapezoidal rule gives $16.3350$ and Simpson's rule gives 16.3333.
E) The Trapezoidal rule gives $16.3350$ and Simpson's rule gives 17.1518.

Question

Quizplus · Accepted Answer

Using the Trapezoidal Rule with 20 subintervals, we have
\begin{align*}
\int _ { 1 } ^ { 2 } \left( 4 z ^ { 2 } + 7 ight) d z &\approx \frac{1}{2}(f(1)+f(1.1))+f(1.2)+f(1.3)+\cdots+f(1.9)+f(2)\
&\approx \frac{0.1}{2}\left[(4\cdot1^2+7)+(4\cdot1.1^2+7)ight]\
&\quad+(4\cdot1.2^2+7)+(4\cdot1.3^2+7)+\cdots+(4\cdot1.9^2+7)\
&\quad+(4\cdot2^2+7)\
&\approx 16.3350
\end{align*}
Using Simpson's Rule with 20 subintervals, we have
\begin{align*}
\int _ { 1 } ^ { 2 } \left( 4 z ^ { 2 } + 7 ight) d z &\approx \frac{1}{3h}\left[f(1)+4f(1.5)+2f(2)+4f(1.25)+\cdots+2f(1.9)+4f(1.95)+f(2)ight]\
&\approx \frac{1}{3(0.05)}\left[(4\cdot1^2+7)+4(4\cdot1.5^2+7)+2(4\cdot2^2+7)ight]\
&\quad+4(4\cdot1.25^2+7)+(4\cdot1.3^2+7)+\cdots+2(4\cdot1.95^2+7)+(4\cdot2^2+7)\
&\approx 16.3333
\end{align*}
The exact value of the definite integral is
\begin{align*}
\int _ { 1 } ^ { 2 } \left( 4 z ^ { 2 } + 7 ight) d z &= \left[ \frac { 4 z ^ { 3 } } { 3 } + 7 z ight] _ { 1 } ^ { 2 }\
&= \frac{32}{3}+14\
&= \frac{74}{3}\
&\approx 24.6667
\end{align*}
The Trapezoidal rule gives a more accurate approximation, so the answer is D.

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