Questions: Calcular el área de la región sombreada. A) 152/3 B) 165/4 C) 625/12 D) 128/3

Transcript text: Calcular el área de la región sombreada. A) $\frac{152}{3}$ B) $\frac{165}{4}$ C) $\frac{625}{12}$ D) $\frac{128}{3}$

Solution

Solution Steps

Step 1: Identify the function and the region to be integrated

The function given is $ f(x) = x(x-5)^2 $. The shaded region under the curve from $ x = 0 $ to $ x = 5 $ needs to be calculated.

Step 2: Set up the integral

To find the area under the curve, we need to integrate the function $ f(x) $ from 0 to 5: \[ \int_{0}^{5} x(x-5)^2 \, dx \]

Step 3: Expand the integrand

Expand the integrand $ x(x-5)^2 $: \[ x(x-5)^2 = x(x^2 - 10x + 25) = x^3 - 10x^2 + 25x \]

Step 4: Integrate the expanded function

Integrate each term separately: \[ \int_{0}^{5} (x^3 - 10x^2 + 25x) \, dx = \int_{0}^{5} x^3 \, dx - 10 \int_{0}^{5} x^2 \, dx + 25 \int_{0}^{5} x \, dx \]

Step 5: Compute each integral

Calculate each integral: \[ \int_{0}^{5} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{0}^{5} = \frac{5^4}{4} - \frac{0^4}{4} = \frac{625}{4} \] \[ \int_{0}^{5} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{5} = \frac{5^3}{3} - \frac{0^3}{3} = \frac{125}{3} \] \[ \int_{0}^{5} x \, dx = \left[ \frac{x^2}{2} \right]_{0}^{5} = \frac{5^2}{2} - \frac{0^2}{2} = \frac{25}{2} \]

Step 6: Combine the results

Combine the results of the integrals: \[ \int_{0}^{5} (x^3 - 10x^2 + 25x) \, dx = \frac{625}{4} - 10 \left( \frac{125}{3} \right) + 25 \left( \frac{25}{2} \right) \] \[ = \frac{625}{4} - \frac{1250}{3} + \frac{625}{2} \]

Step 7: Find a common denominator and simplify

Find a common denominator (12) and simplify: \[ = \frac{625 \cdot 3}{12} - \frac{1250 \cdot 4}{12} + \frac{625 \cdot 6}{12} \] \[ = \frac{1875}{12} - \frac{5000}{12} + \frac{3750}{12} \] \[ = \frac{1875 - 5000 + 3750}{12} \] \[ = \frac{625}{12} \]