Question
3 Multiple Choice 1 point Find the area between the curves \( y=x^{2} \) and \( y=x^{3} \)
Ask by Boone Owen. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The area between the curves \( y = x^2 \) and \( y = x^3 \) is \(\frac{1}{12}\).
Solution
To find the area between the curves \( y = x^2 \) and \( y = x^3 \), follow these steps:
1. **Find the Points of Intersection:**
Set \( x^2 = x^3 \):
\[
x^2(1 - x) = 0 \implies x = 0 \text{ or } x = 1
\]
So, the curves intersect at \( x = 0 \) and \( x = 1 \).
2. **Determine Which Function Is on Top:**
For \( 0 < x < 1 \):
\[
x^2 > x^3
\]
Thus, \( y = x^2 \) is above \( y = x^3 \) in this interval.
3. **Calculate the Area:**
\[
\text{Area} = \int_{0}^{1} (x^2 - x^3) \, dx = \left[ \frac{x^3}{3} - \frac{x^4}{4} \right]_0^1 = \left( \frac{1}{3} - \frac{1}{4} \right) = \frac{1}{12}
\]
**Answer:** The area between the curves is \(\frac{1}{12}\).
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Simplify this solution Extra Insights
To find the area between the curves \( y = x^2 \) and \( y = x^3 \), you first need to determine the points where they intersect. Set \( x^2 = x^3 \), which simplifies to \( x^2(1 - x) = 0 \). The solutions are \( x = 0 \) and \( x = 1 \). Now, to find the area between these two curves from \( x = 0 \) to \( x = 1 \), you set up the integral: \[ \text{Area} = \int_0^1 (x^2 - x^3) \, dx \] Calculating this integral gives: \[ = \left[ \frac{x^3}{3} - \frac{x^4}{4} \right]_0^1 = \left( \frac{1}{3} - \frac{1}{4} \right) = \frac{4}{12} - \frac{3}{12} = \frac{1}{12} \] So, the area between the curves is \( \frac{1}{12} \) square units.
