Question

Use a symbolic integration utility to evaluate the double integral. $$ \int_{0}^{3} \int_{0}^{x^{2}} \sqrt{x} \sqrt{1+x} d y d x $$

Short answer

Upon evaluating, the answer to the double integral is: \( \frac{2}{7}(4^{7/2}-1) \)

Step-by-step solution

Integration with respect to y

As the integrand does not contain the variable \(y\), we first integrate with respect to \(y\). The integral of a constant with respect to a variable simply becomes that constant times the variable. So, we have: \[\int_{0}^{x^{2}} \sqrt{x} \sqrt{1+x} dy = \sqrt{x} \sqrt{1+x} \times \int_{0}^{x^{2}} dy = \sqrt{x} \sqrt{1+x} \times [y]_{0}^{x^{2}} = \sqrt{x} \sqrt{1+x} \times x^{2}\]

Simplify the expression

After simplifying the expression, we get: \[ \sqrt{x} \sqrt{1+x} \times x^{2} = x^{2}x^{1/2}(1+x)^{1/2} = x^{5/2}(1+x)^{1/2}\]

Integration with respect to x

Now, with the new expression, we integrate with respect to \(x\) from 0 to 3. This would require knowledge of integral techniques, and in this case, the use of substitution method: \[ \int_{0}^{3} x^{5/2} (1+x)^{1/2} dx \] The use of a substitution method to compute this integral would be ideal. A possible substitution is \(u = 1+x\). Calculate \(du/dx\), set dx, substitute these values into the equation, and proceed to find the integral. Thus: \[\int_{0}^{3} x^{5/2} (1+x)^{1/2} dx = \frac{2}{7}(u^{7/2}-1)_{1}^{4}\]