To evaluate the integral $\int_{1}^{4} \left(5x - 5\sqrt{x}\right) \, dx$ using the Fundamental Theorem of Calculus, let's break down each term and solve the integral step-by-step.

Step 1: Set up the Integral

The integral is: $\int_{1}^{4} \left(5x - 5\sqrt{x}\right) \, dx$

Step 2: Separate the Terms

We can split the integral: $\int_{1}^{4} 5x \, dx - \int_{1}^{4} 5\sqrt{x} \, dx$

Step 3: Integrate Each Term

Integrate $5x$:

$\int 5x \, dx = \frac{5x^2}{2}$ Evaluated from $1$ to $4$: [ \left[\frac{5 \cdot 4^2}{2}\right] - \left[\frac{5 \cdot 1^2}{2}\right] = \left[\frac{5 \cdot 16}{2}\right] - \left[\frac{5 \cdot 1}{2}\right] = 40 - 2.5 = 37.5 ]

Integrate $5\sqrt{x}$:

Rewrite $\sqrt{x}$ as $x^{1/2}$: $\int 5\sqrt{x} \, dx = \int 5x^{1/2} \, dx = 5 \cdot \frac{2}{3} x^{3/2} = \frac{10}{3} x^{3/2}$ Evaluated from $1$ to $4$: [ \left[\frac{10}{3} \cdot 4^{3/2}\right] - \left[\frac{10}{3} \cdot 1^{3/2}\right] = \left[\frac{10}{3} \cdot 8\right] - \left[\frac{10}{3} \cdot 1\right] = \frac{80}{3} - \frac{10}{3} = \frac{70}{3} ]

Step 4: Combine Results

Now, subtract the results from each integral: $37.5 - \frac{70}{3} = \frac{112.5 - 70}{3} = \frac{42.5}{3} = 14.1667$

Final Answer

$\int_{1}^{4} \left(5x - 5\sqrt{x}\right) \, dx = 14.1667$

Would you like a graph of the integrand or further details on any part of the solution?

Related Questions:

1. How do you apply the Fundamental Theorem of Calculus to more complex integrands?
2. What are some techniques to integrate functions with roots or fractional exponents?
3. How can substitution simplify integrals like $\int \sqrt{x} \, dx$?
4. What is the significance of the shaded area in integral calculations?
5. How does changing the limits of integration affect the net area calculated?

Tip: Breaking down integrals by separating terms often simplifies complex expressions and aids in solving each part individually.