Question

the value of the integral \\(\int_{0}^{4.2} s^2 \\, ds = \square\\). (type an exact answer in simplified form.)

Explanation:

Step1: Recall the power rule for integration

The power rule for integration states that $\int s^n ds = \frac{s^{n + 1}}{n + 1} + C$ (where $C$ is the constant of integration) for $n
eq - 1$. For the integral $\int_{0}^{4.2}s^{2}ds$, we use this rule with $n = 2$.
So, $\int s^{2}ds=\frac{s^{3}}{3}+C$.

Step2: Evaluate the definite integral

To evaluate the definite integral $\int_{a}^{b}f(s)ds=F(b)-F(a)$, where $F(s)$ is the antiderivative of $f(s)$. Here, $f(s)=s^{2}$, $F(s)=\frac{s^{3}}{3}$, $a = 0$ and $b = 4.2$.
First, substitute $s = 4.2$ into $F(s)$: $F(4.2)=\frac{(4.2)^{3}}{3}$.
Then substitute $s = 0$ into $F(s)$: $F(0)=\frac{0^{3}}{3}=0$.
Now, subtract: $\int_{0}^{4.2}s^{2}ds=F(4.2)-F(0)=\frac{(4.2)^{3}}{3}-0$.
Calculate $(4.2)^{3}=4.2\times4.2\times4.2 = 74.088$.
Then $\frac{74.088}{3}=24.696$. But we can also write $4.2=\frac{21}{5}$, so $(4.2)^{3}=(\frac{21}{5})^{3}=\frac{9261}{125}$. Then $\frac{(4.2)^{3}}{3}=\frac{9261}{125\times3}=\frac{3087}{125}=24.696$.

Answer:

24.696 (or $\frac{3087}{125}$)