What is the final exact value obtained for the displayed arc-length integral?

Conditions

• The integral has been fully substituted and evaluated up to the step $\frac{8}{27}\cdot 26$.
• The arithmetic simplification $8 \cdot 26 = 208$ is performed exactly.

Reasoning, step by step

1. Start with the expression $\frac{8}{27}\cdot 26$.
2. Multiply the numerator 8 by 26. The speaker splits this into $8 \cdot 20 = 160$ and $8 \cdot 6 = 48$.
3. Add the partial products: $160 + 48 = 208$.
4. Keep the denominator as 27.
5. Write the final exact rational value as $\frac{208}{27}$.

Example

The speaker concludes, "So it's 208 over 27. And we are done." The board writes $=\frac{208}{27}$ and boxes it to mark the end of the worked example.

Common misconceptions

• Converting the exact fraction $\frac{208}{27}$ into a decimal approximation prematurely.
• Making an arithmetic error when multiplying 8 by 26.
• Forgetting to include the denominator 27 in the final answer.

Watch the explanation

Explore next

• How does the fundamental theorem of calculus justify evaluating the definite integral this way?
• What are the conditions required for the arc length formula to be valid?