Find the work done in stretching a spring if force is given by F = 5x, from x = 0 to x = 2.

$V = \pi \int_a^b [f(x)]^2 \, dx$.
Here, $f(x) = x^2$, $a = 0$, and $b = 1$.

$$V = \pi \int_0^1 (x^2)^2 \, dx = \pi \int_0^1 x^4 \, dx$$

$$V = \pi \left(\frac{1}{5} - 0\right) = \frac{\pi}{5}$$

The final answer is $\boxed{\frac{\pi}{5} \text{ cubic units}}$.

Q9. Find the work done in stretching a spring if force is given by $F = 5x$, from $x = 0$ to $x = 2$.

$$W = \int_a^b F(x) \, dx$$

Here, $F(x) = 5x$, $a = 0$, and $b = 2$.

$$W = \int_0^2 5x \, dx$$

$$W = \left(\frac{5}{2} \cdot 4\right) - 0 = 10$$

The final answer is $\boxed{10 \text{ units of work}}$.

Q10. Find the work done by a force $F = 3x^2$ from $x = 1$ to $x = 3$.

$$W = \int_a^b F(x) \, dx$$

Here, $F(x) = 3x^2$, $a = 1$, and $b = 3$.

$$W = \int_1^3 3x^2 \, dx$$

$$W = 27 - 1 = 26$$

The final answer is $\boxed{26 \text{ units of work}}$.

Q11. Find the average value of $f(x) = x^2$ on $[0,3]$.

$f_{avg} = \frac{1}{b-a} \int_a^b f(x) \, dx$.
Here, $f(x) = x^2$, $a = 0$, and $b = 3$.

$$f_{avg} = \frac{1}{3-0} \int_0^3 x^2 \, dx = \frac{1}{3} \int_0^3 x^2 \, dx$$

$$f_{avg} = \frac{1}{3} \cdot 9 = 3$$

The final answer is $\boxed{3}$.