The integral that represents the area of the shaded region is incorrect in the provided solution. Let's analyze and correct it.

Steps to Correct:

1. Understand the boundaries:

   • The integral is evaluated from $x = 1$ to $x = 8$.
   • The top function is $y = \sqrt[3]{x}$.
   • The bottom function is $y = 2 - x$.

2. Set up the integral: The area is calculated as: $\text{Area} = \int_1^8 \left(\text{Top Function} - \text{Bottom Function}\right) dx$ Thus: $\text{Area} = \int_1^8 \left( \sqrt[3]{x} - (2 - x) \right) dx$ Simplify the integrand: $\text{Area} = \int_1^8 \left( \sqrt[3]{x} + x - 2 \right) dx$

3. Compute the integral: Break it into separate terms: [ \int_1^8 \left( \sqrt[3]{x} + x - 2 \right) dx = \int_1^8 \sqrt[3]{x} dx + \int_1^8 x dx - \int_1^8 2 dx ] Evaluate each term:

   • For $\int_1^8 \sqrt[3]{x} dx = \int_1^8 x^{1/3} dx$: $\int x^{1/3} dx = \frac{3}{4}x^{4/3} \Big|_1^8 = \frac{3}{4}(8^{4/3}) - \frac{3}{4}(1^{4/3})$ Compute powers: $8^{4/3} = (8^{1/3})^4 = 2^4 = 16, \quad 1^{4/3} = 1$ Substitute: $\frac{3}{4}(16) - \frac{3}{4}(1) = 12 - \frac{3}{4} = \frac{45}{4}$
   • For $\int_1^8 x dx$: $\int x dx = \frac{x^2}{2} \Big|_1^8 = \frac{8^2}{2} - \frac{1^2}{2} = 32 - 0.5 = 31.5$
   • For $\int_1^8 2 dx$: $\int 2 dx = 2x \Big|_1^8 = 2(8) - 2(1) = 16 - 2 = 14$

4. Combine results: Add up all terms: $\text{Area} = \frac{45}{4} + 31.5 - 14$ Simplify: $\text{Area} = \frac{45}{4} + \frac{63}{2} - 14 = \frac{45}{4} + \frac{126}{4} - \frac{56}{4} = \frac{115}{4} = 28.75$

Final Answer:

The correct integral is: $\int_1^8 \left( \sqrt[3]{x} + x - 2 \right) dx$ And the area is: $28.75$

If you have questions or need further clarifications, let me know!

Here are five follow-up questions:

1. How do you compute integrals with fractional powers of $x$?
2. Why is the area represented by the integral of the difference between two functions?
3. How can we check the correctness of an integral setup graphically?
4. What are the conditions for using definite integrals to compute areas?
5. How can this integral be computed using numerical methods?

Tip: Always verify the limits of integration and the correct functions for top and bottom to avoid mistakes!