Find the double integral of f (x, y) = (x^7)y over the region between the curves y = x^2 and y = x(3 - x).
To find the double integral of \(f(x, y) = x^7 \cdot y\) over the region between the curves \(y = x^2\) and \(y = x(3 - x)\), we can use the double integration technique.
Step 1: Determine the limits of integration:
To find the limits of integration, we need to identify the intersection points of the two curves: \(y = x^2\) and \(y = x(3 - x)\).
Setting the equations equal to each other:
\(x^2 = x(3 - x)\)
Expand and simplify:
\(x^2 = 3x - x^2\)
\(2x^2 - 3x = 0\)
Factoring out common factors:
\(x(2x - 3) = 0\)
Thus, \(x = 0\) or \(x = \frac{3}{2}\).
So the limits of integration for \(x\) will be from 0 to \(\frac{3}{2}\).
For \(y\), we need to find the limits of integration based on the given curves.
The curve \(y = x^2\) has limits of integration for \(y\) from 0 to \(x^2\).
The curve \(y = x(3 - x)\) has limits of integration for \(y\) from \(x(3 - x)\) to \(\frac{3}{2}\).
Hence, the limits of integration for \(y\) will be from \(x(3 - x)\) to \(x^2\).
Step 2: Set up the double integral:
The double integral of \(f(x, y)\) over the given region can be expressed as:
\(\int\int_R x^7 \cdot y \, dy \, dx\)
where \(R\) represents the region between the curves \(y = x^2\) and \(y = x(3 - x)\).
Step 3: Evaluate the double integral:
Now, evaluate the double integral using the limits of integration determined in step 1:
\(\int_0^{\frac{3}{2}} \int_{x(3 - x)}^{x^2} x^7 \cdot y \, dy \, dx\)
By integrating with respect to \(y\) and then \(x\), you can solve this double integral to find the result.