Question

calculate \\( \int_{8}^{5} 4 x^{2} d x \\), given the following.
\\( \int_{4}^{5} x d x=4.5 \quad \int_{4}^{5} x^{2} d x=\frac{61}{3} \quad \int_{5}^{8} x^{2} d x=129 \\)
\\( \int_{8}^{5} 4 x^{2} d x= \\) (type an integer or a simplified fraction.)

Explanation:

Step1: Use the property of definite integrals

We know that \(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\). So \(\int_{8}^{5}4x^{2}dx = 4\int_{8}^{5}x^{2}dx=- 4\int_{5}^{8}x^{2}dx\)

Step2: Calculate the value

Since \(\int_{5}^{8}x^{2}dx = 129\), then \(-4\int_{5}^{8}x^{2}dx=-4\times129\)

Answer:

\(-516\)