QuestionLet $f(x)=\sqrt{x^{2}+x}$ for $1 \leq x \leq 7$. We wish to estimate $$\int_{1}^{7} f(x) d x$$ by the Trapezoidal Rule. a) Divide the domain of $f$ into 4 sub-intervals of equal length. Calculate their common length $\Delta x$ (exact value). b) Find the approximation of $$\int_{1}^{7} f(x) d x$$ that the Trapezoidal Rule produces with 4 sub-intervals. Give the answer with $\pm 0.0001$ precision. Number

Studdy Solution
Calculate the approximation to the specified precision:
First, compute each term:
$\sqrt{2} \approx 1.4142$
$2\sqrt{8.75} \approx 2 \times 2.958 \approx 5.916$
$2\sqrt{20} \approx 2 \times 4.472 \approx 8.944$
$2\sqrt{35.75} \approx 2 \times 5.979 \approx 11.958$
$\sqrt{56} \approx 7.4833$
Now substitute these into the formula:
$\int_{1}^{7} f(x) \, dx \approx \frac{1.5}{2} \left[ 1.4142 + 5.916 + 8.944 + 11.958 + 7.4833 \right]$
Calculate the sum inside the brackets:
$1.4142 + 5.916 + 8.944 + 11.958 + 7.4833 \approx 35.7155$
Now calculate the final approximation:
$\int_{1}^{7} f(x) \, dx \approx \frac{1.5}{2} \times 35.7155 \approx 26.786625$
Rounding to $\pm 0.0001$ precision:
$\boxed{26.7866}$