Question

calculate \\( \int_{7}^{6} 9 x^{2} d x \\), given the following.
\\( \int_{3}^{6} x d x=13.5 \quad \int_{3}^{6} x^{2} d x=63 \quad \int_{6}^{7} x^{2} d x=\frac{127}{3} \\)
\\( \int_{7}^{6} 9 x^{2} d x= \\) (type an integer or a simplified fraction.)

Explanation:

Step1: Use the property of definite integrals \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\)

We know that \(\int_{7}^{6}9x^{2}dx = 9\int_{7}^{6}x^{2}dx=- 9\int_{6}^{7}x^{2}dx\)

Step2: Substitute the given value of \(\int_{6}^{7}x^{2}dx\)

Since \(\int_{6}^{7}x^{2}dx=\frac{127}{3}\), then \(9\int_{7}^{6}x^{2}dx=-9\times\frac{127}{3}\)

Step3: Simplify the expression

\(-9\times\frac{127}{3}=- 3\times127=-381\)

Answer:

\(-381\)