We are given the double integral:

$I = \int_0^a \int_0^{\frac{b}{\sqrt{a^2 - x^2}}} x^2 \, dy \, dx$

Step 1: Understanding the Region of Integration

The limits describe a region where:

• $x$ varies from $0$ to $a$.
• $y$ varies from $0$ to $\frac{b}{\sqrt{a^2 - x^2}}$.

To change the order of integration, we express the region in terms of $y$ first.
From the upper limit of $y$, we solve for $x$:

$y = \frac{b}{\sqrt{a^2 - x^2}}$

Squaring both sides:

$y^2 (a^2 - x^2) = b^2$

Rearrange:

$x^2 = a^2 - \frac{b^2}{y^2}$

So, for a fixed $y$, $x$ varies from $0$ to $\sqrt{a^2 - \frac{b^2}{y^2}}$. The value of $y$ starts from $0$ and extends to its maximum when $x = 0$, which gives $y = \frac{b}{a}$.

Step 2: Writing the Integral with Changed Order

The new limits are:

• $y$ varies from $0$ to $\frac{b}{a}$.
• $x$ varies from $0$ to $\sqrt{a^2 - \frac{b^2}{y^2}}$.

Thus, the integral becomes:

$I = \int_0^{\frac{b}{a}} \int_0^{\sqrt{a^2 - \frac{b^2}{y^2}}} x^2 \, dx \, dy$

Step 3: Evaluating the Inner Integral

We compute:

$\int_0^{\sqrt{a^2 - \frac{b^2}{y^2}}} x^2 \, dx$

Using the standard formula:

$\int x^2 \, dx = \frac{x^3}{3}$

Evaluating from $0$ to $\sqrt{a^2 - \frac{b^2}{y^2}}$:

$\frac{1}{3} \left( a^2 - \frac{b^2}{y^2} \right)^{\frac{3}{2}}$

Thus, our integral reduces to:

$I = \int_0^{\frac{b}{a}} \frac{1}{3} \left( a^2 - \frac{b^2}{y^2} \right)^{\frac{3}{2}} dy$

Step 4: Substituting $z = \frac{b}{y}$

Define $z = \frac{b}{y}$, so $dz = -\frac{b}{y^2} dy$, or equivalently:

$dy = -\frac{b}{z^2} dz$

When $y = 0$, $z \to \infty$, and when $y = \frac{b}{a}$, $z = a$. Thus, the limits become $z$ from $\infty$ to $a$, and our integral transforms into:

$I = \int_{\infty}^{a} \frac{1}{3} \left( a^2 - z^2 \right)^{\frac{3}{2}} \cdot \left( -\frac{b}{z^2} \right) dz$

Since the negative sign flips the limits, we write:

$I = \frac{b}{3} \int_a^{\infty} \frac{\left( a^2 - z^2 \right)^{\frac{3}{2}}}{z^2} dz$

This integral is more challenging and requires special functions (such as Beta functions) to evaluate explicitly.

Would you like a numerical approach or further simplifications?

Relative Questions

1. How do we interpret changing the order of integration geometrically?
2. Can we solve this integral using polar coordinates?
3. What are some techniques to evaluate difficult integrals involving square roots?
4. How does substitution help in integral evaluation?
5. How do Beta and Gamma functions relate to this integral?

Tip

When changing the order of integration, always sketch the region first to understand the new limits properly.