https://file.aiquickdraw.com/mathbot/file/1725635837915pjzg0kmd.jpg

To find the area under the curve \( y = x^3 \) from \( x = 0 \) to \( x = 3 \), we will use definite integration. The area under the curve is given by the integral:

\[
A = \int_0^3 x^3 \, dx
\]

Now, let's compute the integral step by step:

1. Find the indefinite integral of \( x^3 \):
\[
\int x^3 \, dx = \frac{x^4}{4}
\]

2. Apply the limits of integration from 0 to 3:
\[
A = \left[ \frac{x^4}{4} \right]_0^3 = \frac{3^4}{4} - \frac{0^4}{4} = \frac{81}{4} - 0 = \frac{81}{4} = 20.25
\]

Thus, the area under the curve \( y = x^3 \) from \( x = 0 \) to \( x = 3 \) is \( 20.25 \) square units.

Would you like any further details or have any questions?

Here are five related questions you might find interesting:
1. How do you compute the area under a curve for a more complex function?
2. What is the geometric interpretation of definite integrals?
3. How does the integral change if the limits of integration are reversed?
4. How do you solve the area problem if the curve crosses the x-axis?
5. How would you find the area between two curves?

**Tip:** Definite integrals not only help find areas but also can be applied to solve problems involving displacement, work, and other physical quantities.

Find the area under the curve y = x^3 for values between 0 and 3.

This page explains how to find the area under the curve y = x^3 between the limits 0 and 3 using definite integration. Step-by-step explanation of the integral computation, including the use of the formula for the area under the curve. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation