Find the area of the shaded region- y1= x=10y^3, y2= x=10y^2, and a plotted point at (10,1)

To find the area of the shaded region between the curves \( y_1 = x = 10y^3 \) and \( y_2 = x = 10y^2 \), we can follow these steps:

### 1. **Express the equations in terms of \( y \):**

Given:
\[
x = 10y^3 \quad \text{(Curve 1)}
\]
\[
x = 10y^2 \quad \text{(Curve 2)}
\]

### 2. **Determine the points of intersection:**

To find the points where the curves intersect, set the equations equal to each other:
\[
10y^3 = 10y^2
\]

Divide both sides by 10:
\[
y^3 = y^2
\]

Factor out \( y^2 \):
\[
y^2(y - 1) = 0
\]

This gives us:
\[
y = 0 \quad \text{or} \quad y = 1
\]

These are the \( y \)-coordinates of the intersection points. We also know the plotted point at \((10, 1)\) confirms \( y = 1 \) as an intersection point.

### 3. **Determine the corresponding \( x \)-coordinates:**

For \( y = 0 \):
\[
x = 10(0)^3 = 0
\]

For \( y = 1 \):
\[
x = 10(1)^3 = 10
\]

So the points of intersection are \( (0, 0) \) and \( (10, 1) \).

### 4. **Set up the integral to find the area:**

The area between the curves can be found by integrating the difference between the rightmost and leftmost functions with respect to \( y \):
\[
\text{Area} = \int_{y=0}^{y=1} \left[ (10y^2) - (10y^3) \right] \, dy
\]

### 5. **Compute the integral:**

\[
\text{Area} = \int_{0}^{1} \left(10y^2 - 10y^3\right) \, dy
\]

Integrating term by term:
\[
\text{Area} = 10\left[\frac{y^3}{3}\right]_{0}^{1} - 10\left[\frac{y^4}{4}\right]_{0}^{1}
\]

Evaluate this:
\[
\text{Area} = 10\left(\frac{1^3}{3} - \frac{0^3}{3}\right) - 10\left(\frac{1^4}{4} - \frac{0^4}{4}\right)
\]
\[
\text{Area} = 10\left(\frac{1}{3}\right) - 10\left(\frac{1}{4}\right)
\]
\[
\text{Area} = \frac{10}{3} - \frac{10}{4}
\]
\[
\text{Area} = \frac{40}{12} - \frac{30}{12}
\]
\[
\text{Area} = \frac{10}{12} = \frac{5}{6}
\]

So, the area of the shaded region is \( \frac{5}{6} \) square units.

Would you like more details, or do you have any other questions?

### Related Questions:
1. How do you find the area between two curves if both are functions of \( x \) instead of \( y \)?
2. What are the conditions under which the integral method can be used to find the area between curves?
3. How would the solution change if the curves were described by parametric equations?
4. What are some real-world applications of finding areas between curves?
5. How does the choice of the variable of integration (x vs y) affect the calculation of areas?

### Tip:
Always check the limits of integration by finding the points of intersection first; this ensures the area calculation is correct.

Learn how to find the area of the shaded region between the curves x = 10y^3 and x = 10y^2, using integration. This example includes step-by-step instructions on finding intersection points and setting up the integral for calculation. Suitable for students in Grades 11-12.

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