To find the area under the curve for the given function $x^4$ from $x = 1$ to $x = 3$, we need to compute the definite integral:

$\int_1^3 x^4 \, dx$

Solution:

1. First, find the antiderivative of $x^4$. The integral of $x^n$ is given by:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$

For $n = 4$, we get:

$\int x^4 \, dx = \frac{x^5}{5}$

2. Now apply the limits of integration, from 1 to 3:

$\left[ \frac{x^5}{5} \right]_1^3 = \frac{3^5}{5} - \frac{1^5}{5}$

3. Compute the values:

$3^5 = 243, \quad 1^5 = 1$

Substitute these into the equation:

$\frac{243}{5} - \frac{1}{5} = \frac{243 - 1}{5} = \frac{242}{5}$

Thus, the area under the curve is:

$\boxed{\frac{242}{5}} \text{ or } 48.4$

Would you like further details or have any questions about this solution?

Related Questions:

1. How do you compute the area under a curve for more complex functions?
2. What is the geometric meaning of a definite integral?
3. How do improper integrals differ from definite integrals?
4. Can integrals be used to find the area between two curves?
5. How can we apply integration techniques to solve physics problems?

Tip:

Always double-check limits of integration and ensure you calculate the antiderivative correctly before substituting the limits!