find the total area of the shaded region.\n1.5\n1\nx = 5y³\n(5,1)\nx = 5y²\n0.5\n0\n2.5\n5\n7.5

Explanation:

Step 1: Determine the interval and integrand

The curves intersect at ( y = 0 ) and ( y = 1 ). The right curve is ( x = 5y^2 ), the left curve is ( x = 5y^3 ). The area between them is the integral of their difference. $$ \text{Area} = \int_{0}^{1} \left(5y^2 - 5y^3\right) dy $$

Step 2: Integrate term by term

Integrate each part separately using the power rule ( \int y^n dy = \frac{y^{n+1}}{n+1} ). $$ \int 5y^2 dy = 5 \cdot \frac{y^3}{3}, \quad \int 5y^3 dy = 5 \cdot \frac{y^4}{4} $$

Step 3: Evaluate from 0 to 1

Substitute the upper limit ( y = 1 ) and subtract the lower limit ( y = 0 ) (which yields 0). $$ \left[ \frac{5y^3}{3} - \frac{5y^4}{4} \right]_{0}^{1} = \left( \frac{5}{3} - \frac{5}{4} \right) - 0 $$

Step 4: Simplify the result

Find a common denominator and subtract. $$ \frac{5}{3} - \frac{5}{4} = \frac{20 - 15}{12} = \frac{5}{12} $$

Answer:

$\frac{5}{12}$