Let's solve the problem step-by-step: **Given:** The curve is \( y = 10x^{1/2} - \frac{5}{2}x^{3/2} \) It meets the x-axis at \( (0,0) \) and \( (4,0) \). We are asked to find the area of the shaded region under the curve from \( x = 0 \) to \( x = 4 \). --- ## **Step 1: Write the Area as an Integral** The area under the curve from \( x = 0 \) to \( x = 4 \) is: \[ A = \int_{0}^{4} \left(10x^{1/2} - \frac{5}{2}x^{3/2}\right) dx \] --- ## **Step 2: Integrate Each Term** First, let's integrate \( 10x^{1/2} \): \[ \int 10x^{1/2} dx = 10 \int x^{1/2} dx = 10 \cdot \frac{2}{3}x^{3/2} = \frac{20}{3}x^{3/2} \] Now, integrate \( -\frac{5}{2}x^{3/2} \): \[ \int -\frac{5}{2}x^{3/2} dx = -\frac{5}{2} \int x^{3/2} dx = -\frac{5}{2} \cdot \frac{2}{5}x^{5/2} = -x^{5/2} \] So, the indefinite integral is: \[ \int \left(10x^{1/2} - \frac{5}{2}x^{3/2}\right) dx = \frac{20}{3}x^{3/2} - x^{5/2} \] --- ## **Step 3: Apply the Limits** Now evaluate from \( x = 0 \) to \( x = 4 \): \[ A = \left[ \frac{20}{3}x^{3/2} - x^{5/2} \right]_{0}^{4} \] Calculate at \( x = 4 \): - \( 4^{3/2} = (4^1)^{3/2} = (2^2)^{3/2} = 2^{3} = 8 \) - \( 4^{5/2} = (4^1)^{5/2} = (2^2)^{5/2} = 2^{5} = 32 \) Plug in \( x = 4 \): \[ \frac{20}{3} \times 8 - 32 = \frac{160}{3} - 32 \] At \( x = 0 \): \[ \frac{20}{3} \times 0 - 0 = 0 \] So, \[ A = \left(\frac{160}{3} - 32\right) - 0 = \frac{160}{3} - 32 \] --- ## **Step 4: Simplify the Answer** Convert 32 to thirds: \[ 32 = \frac{96}{3} \] So, \[ A = \frac{160}{3} - \frac{96}{3} = \frac{64}{3} \] --- ## **Final Answer** \[ \boxed{\frac{64}{3}} \] The area of the shaded region is **\(\frac{64}{3}\) square units**.