https://file.aiquickdraw.com/mathbot/file/17265689676223youomie.jpg

The problem in the image is asking to find the area between the curves:

\[
y = x^2 \quad \text{and} \quad y = 2x - x^2
\]

### Steps to solve:

1. **Find the points of intersection**: Set the two equations equal to each other to find where the curves intersect.

\[
x^2 = 2x - x^2
\]

Simplifying this:

\[
x^2 + x^2 = 2x \quad \Rightarrow \quad 2x^2 - 2x = 0 \quad \Rightarrow \quad 2x(x - 1) = 0
\]

So, the points of intersection are \(x = 0\) and \(x = 1\).

2. **Set up the integral**: The area between two curves is given by the integral of the difference between the upper and lower functions over the interval of intersection. In this case, \(y = 2x - x^2\) is the upper curve and \(y = x^2\) is the lower curve between \(x = 0\) and \(x = 1\).

\[
\text{Area} = \int_0^1 [(2x - x^2) - x^2] \, dx = \int_0^1 (2x - 2x^2) \, dx
\]

3. **Compute the integral**:

\[
\int_0^1 (2x - 2x^2) \, dx = \left[ x^2 - \frac{2x^3}{3} \right]_0^1
\]

Evaluate the integral:

\[
= \left[ 1^2 - \frac{2(1^3)}{3} \right] - \left[ 0^2 - \frac{2(0^3)}{3} \right] = \left( 1 - \frac{2}{3} \right) = \frac{1}{3}
\]

Thus, the area between the curves is \(\frac{1}{3}\) square units.

Would you like more details on any step, or have further questions?

Here are some related questions you might find helpful:

1. How do you find the area between more complex curves?
2. Can you calculate the area if the curves intersect at multiple points?
3. How does the concept of definite integrals apply to finding areas?
4. What if the upper and lower curves change position within the interval?
5. Can you explain why we subtract the lower curve from the upper curve?

**Tip**: Always sketch the curves or visualize them to confirm which function is on top and which is on the bottom across the interval.

Find the area between y = x^2 and y = 2x - x^2.

Learn how to find the area between the curves y = x^2 and y = 2x - x^2. This problem involves calculus concepts such as integration and areas between curves. Suitable for students in Grades 11-12 or Calculus 1.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation