Find the area between the curves \( y=x^{2} \) and \( y=x^{3} \) \( 1 / 6 \) \( 1 / 4 \) \( 1 / 12 \) \( 1 / 3 \)

To find the area between the curves \( y = x^2 \) and \( y = x^3 \), you first need to determine the points of intersection. Set \( x^2 = x^3 \), which gives \( x^2 (1 - x) = 0 \). This results in intersection points at \( x = 0 \) and \( x = 1 \). The area between the curves can be found by integrating the difference \( x^2 - x^3 \) from 0 to 1: \[ \text{Area} = \int_{0}^{1} (x^2 - x^3) \, dx = \int_{0}^{1} x^2 \, dx - \int_{0}^{1} x^3 \, dx \] Calculating the integrals: \[ \int_{0}^{1} x^2 \, dx = \left[\frac{x^3}{3}\right]_{0}^{1} = \frac{1}{3} \] \[ \int_{0}^{1} x^3 \, dx = \left[\frac{x^4}{4}\right]_{0}^{1} = \frac{1}{4} \] So the area is: \[ \text{Area} = \frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12} \] Thus, the area between the curves is \( \frac{1}{12} \). Expressing calculations in practical scenarios often helps students interpret mathematical operations. Consider finding the area between curves in real life; architects often need to calculate areas for design layouts. Imagine calculating the area of shaded regions on blueprints to optimize materials or space! If you're tackling integration problems like this, it’s vital to carefully set up your limits and check your intersection points. A common mistake is forgetting to subtract the lower curve from the upper curve, which could lead to an incorrect area calculation. Also, when setting up integrals, ensure you’re integrating with respect to the correct variable for clarity.