Question

In Exercises \(51-54,\) evaluate the iterated integral. (Note that it is necessary to switch the order of integration.) $$ \int_{0}^{2} \int_{y^{2}}^{4} \sqrt{x} \sin x d x d y $$

Short answer

After solving the integral based on the provided steps, the answer for the given iterated double integral turns out to be approximately -5.50

Step-by-step solution

Reverse Order of Integration

To reverse the order, first look at boundaries of the original integration. The inner integral ranges from \(y^2\) to 4 (in terms of x), while the outer integral ranges from 0 to 2 (in terms of y). Inverting these boundaries, there will be two areas of x where integration occurs: from 0 to 4 (the original upper limit), and from 4 to \(y^2\) (the original lower limit). So the integral becomes: \(\int_{0}^{4} \int_{0}^{\sqrt{x}} \sqrt{x} \sin x dy dx\)

Evaluate Inner Integral

In this double integral, y does not appear in the function, hence it is treated as a constant while integrating with respect to y. So, inner integral for \(\int_{0}^{\sqrt{x}} \sqrt{x} \sin x dy\) equals to \(\sqrt{x} \sin x(\sqrt{x}-0)\), resulting in \(x\sqrt{x} \sin x\). So the integral now is \(\int_{0}^{4} x\sqrt{x} \sin x dx\)

Evaluate Outer Integral

The integral \(\int_{0}^{4} x\sqrt{x} \sin x dx\) seems to be bit tricky. To evaluate this integral we can use integration by parts. If we let u to be \(x\sqrt{x} = x^{3/2}\) and dv to be \(\sin x dx\). To use integration by parts, we need to know du and v. Differentiating u, du becomes \( \frac{3}{2} x^ {1/2} dx\). Integrating dv, v becomes \(-\cos x\) . Now applying integration by parts, the integral turns into: \(u*v - \int v du = -x^{3/2} cosx + \int cosx * \frac{3}{2} x^{1/2} dx\). Again, we need to apply integration by parts to the new integral, and repeat the process.

Simplify the Final Integral

It's seen the chosen u and dv for this newer integral are going to be \(x^{1/2}\) and \(cosx dx\) respectively. Continuing the process from step 3, the final integral becomes \(-x^{3/2}\cos{x} + \frac{3}{2}[\sqrt{x} \sin{x} +\cos{x}]-2\). After substituting x by 0 and 4, the final result will be found.