First slide

Evaluation of definite integrals

Question

Using the fact that $0 \leq f (x) \leq g (x), c < x < d$
$\Rightarrow \int_{c}^{d} f (x) d x \leq \int_{c}^{d} g (x) d x$ we can conclude that
$\int_{1}^{3} \sqrt{3 + x^{3}}$ lies in the interval

Moderate

A

$(\frac{1}{2}, 3)$
B

$(2, \sqrt{30})$
C

$(\frac{3}{2}, 5)$
D

$(4, 2 \sqrt{30})$

Solution

For $x \in (1, 3), 2 < \sqrt{3 + x^{3}} < \sqrt{30}$ so
$\begin{array}{l} 2 \int_{1}^{3} d x < \int_{1}^{3} \sqrt{3 + x^{3}} d x < \sqrt{30} \int_{1}^{3} d x \\ \Rightarrow 4 < \int_{1}^{3} \sqrt{3 + x^{3}} d x < 2 \sqrt{30} \end{array}$

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