Question

calculate \\( \int_{3}^{6} 6 x^{2} d x \\), given the following.
\\( \int_{3}^{5} x^{2} d x=\frac{98}{3} \quad \int_{5}^{6} x^{2} d x=\frac{91}{3} \quad \int_{3}^{5} x d x=8 \\)
\\( \int_{3}^{6} 6 x^{2} d x= \\) (type an integer or a simplified fraction.)

Explanation:

Step1: Use integral property \(\int_{a}^{b}cf(x)dx = c\int_{a}^{b}f(x)dx\)

$$\int_{3}^{6}6x^{2}dx=6\int_{3}^{6}x^{2}dx$$

Step2: Use integral property \(\int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx\) (\(a < b < c\))

$$\int_{3}^{6}x^{2}dx=\int_{3}^{5}x^{2}dx+\int_{5}^{6}x^{2}dx$$

Step3: Substitute the given values

Since \(\int_{3}^{5}x^{2}dx=\frac{98}{3}\) and \(\int_{5}^{6}x^{2}dx = \frac{91}{3}\)
$$\int_{3}^{6}x^{2}dx=\frac{98}{3}+\frac{91}{3}=\frac{98 + 91}{3}=\frac{189}{3}=63$$

Step4: Calculate the original integral

$$\int_{3}^{6}6x^{2}dx=6\times63 = 378$$

Answer:

\(378\)