Question

score:
33.33%,
4 of 12
points
points:
0 of 1
calculate \\( \int_{2}^{9} 3 x^{2} d x \\),
given the following.
\\( \int_{2}^{8} x^{2} d x=168 \\)
\\( \int_{8}^{9} x^{2} d x=\frac{217}{3} \\)
\\( \int_{2}^{8} x d x=30 \\)
\\( \int_{2}^{9} 3 x^{2} d x=\square \\)
(type an integer or a
simplified fraction.)

Explanation:

Step1: Use the property of definite integrals

We know that \(\int_{a}^{b}cf(x)dx = c\int_{a}^{b}f(x)dx\) (where \(c\) is a constant). So, \(\int_{2}^{9}3x^{2}dx=3\int_{2}^{9}x^{2}dx\).

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

We know that \(\int_{2}^{9}x^{2}dx=\int_{2}^{8}x^{2}dx+\int_{8}^{9}x^{2}dx\). Given \(\int_{2}^{8}x^{2}dx = 168\) and \(\int_{8}^{9}x^{2}dx=\frac{217}{3}\).
Then \(\int_{2}^{9}x^{2}dx=168+\frac{217}{3}\).

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Step3: Calculate the original integral

Since \(\int_{2}^{9}3x^{2}dx = 3\int_{2}^{9}x^{2}dx\), substitute \(\int_{2}^{9}x^{2}dx=\frac{721}{3}\) into it.
\(\int_{2}^{9}3x^{2}dx=3\times\frac{721}{3}\)

Answer:

\(721\)