To solve the integral [tex]\(\int_{189}^{875} \sqrt{1+\frac{4}{440}\left(\frac{7}{y}\right)^{2 / 3}} \, dy\)[/tex], we need to follow several steps. Here is the detailed, step-by-step solution:
1. Understand the integrand:
The integrand is given as:
[tex]\[
\sqrt{1+\frac{4}{440}\left(\frac{7}{y}\right)^{2 / 3}}
\][/tex]
2. Simplify the expression inside the square root:
[tex]\[
\frac{4}{440} \left( \frac{7}{y} \right)^{2/3} = \frac{1}{110} \left( \frac{7}{y} \right)^{2/3}
\][/tex]
Therefore, the integrand becomes:
[tex]\[
\sqrt{1 + \frac{1}{110} \left( \frac{7}{y} \right)^{2/3}}
\][/tex]
3. Set up the definite integral:
The integral we need to evaluate is:
[tex]\[
\int_{189}^{875} \sqrt{1 + \frac{1}{110} \left( \frac{7}{y} \right)^{2/3}} \, dy
\][/tex]
4. Perform the integration:
This integral doesn't have a straightforward antiderivative, so it must be evaluated using numerical methods. Approximating the integral numerically will provide the most accurate result.
5. Evaluate the integral from [tex]\(y = 189\)[/tex] to [tex]\(y = 875\)[/tex]:
Using numerical integration techniques (like the ones provided by integration libraries in computative tools), we find that the integral evaluates to:
[tex]\[
686.1908801748295
\][/tex]
6. Estimate the margin of error:
Along with this result, the numerical methods also indicate the margin of error, which is extremely small:
[tex]\[
2.0236685149246077 \times 10^{-11}
\][/tex]
Therefore, the final answer to the integral [tex]\(\int_{189}^{875} \sqrt{1+\frac{4}{440}\left(\frac{7}{y}\right)^{2 / 3}} \, dy\)[/tex] is approximately:
[tex]\[
\boxed{686.1908801748295}
\][/tex]
