1. **Problem Statement:** Find the points of intersection of the curves $y = x^3$ and $x = y^2$. 2. **Step 1: Express both equations clearly:** - Curve 1: $y = x^3$ - Curve 2: $x = y^2$ 3. **Step 2: Substitute $y$ from Curve 1 into Curve 2:** Since $y = x^3$, then $x = (x^3)^2 = x^6$. 4. **Step 3: Solve for $x$:** $$x = x^6 \implies x^6 - x = 0 \implies x(x^5 - 1) = 0$$ 5. **Step 4: Find roots:** - $x = 0$ - $x^5 = 1 \implies x = 1$ 6. **Step 5: Find corresponding $y$ values:** - For $x=0$, $y = 0^3 = 0$ - For $x=1$, $y = 1^3 = 1$ 7. **Step 6: Points of intersection:** $$ (0,0) \text{ and } (1,1) $$ 8. **Step 7: Sketch and representative strip:** - The curves intersect at $(0,0)$ and $(1,1)$. - The area bounded lies between these points. - The representative strip is vertical (along $x$), between the curves $y = x^3$ (lower) and $y = \sqrt{x}$ (since $x = y^2 \Rightarrow y = \sqrt{x}$) (upper). **Final answer:** Points of intersection are $$ (0,0) \text{ and } (1,1) $$.