Change of order for double integration f(x,y) dxdy inner limit [(b/a)√(a^2-x^2)] to b and outer limit c to a, c<a.

We are given a double integral with the limits specified in the following order:

\[
\int_c^a \left( \int_{\frac{b}{a}\sqrt{a^2 - y^2}}^b f(x, y) \, dx \right) dy
\]

Our goal is to change the order of integration. To do this, we need to carefully analyze the limits of the current order and express them in terms of \(y\) instead of \(x\).

### Step 1: Understand the limits of the integration
- The outer integral is from \(y = c\) to \(y = a\).
- The inner integral for each \(y\) is from \(x = \frac{b}{a}\sqrt{a^2 - y^2}\) to \(x = b\).

This tells us that for each fixed \(y\), \(x\) ranges between \( \frac{b}{a}\sqrt{a^2 - y^2} \) and \( b \).

### Step 2: Sketch the region of integration
To change the order, we need to determine how \(x\) and \(y\) vary with each other. Let's analyze the curve \( x = \frac{b}{a}\sqrt{a^2 - y^2} \).

- When \(y = a\), \(x = 0\).
- When \(y = c\), \(x = \frac{b}{a}\sqrt{a^2 - c^2}\).

The upper limit of \(x\) is \(x = b\), which is independent of \(y\). So, we know that \(x\) ranges between \(0\) and \(b\).

### Step 3: Change the order of integration
We now reverse the order of integration. First, we determine the limits of \(y\) for a given \(x\):

- From the equation \(x = \frac{b}{a}\sqrt{a^2 - y^2}\), we can solve for \(y\):

\[
y = \sqrt{a^2 - \left(\frac{a}{b}x\right)^2}
\]

Thus, \(y\) ranges from \(c\) to \(a\), and for each \(x\), \(y\) varies between \( \sqrt{a^2 - \left(\frac{a}{b}x\right)^2} \) and some constant values.

The new double integral in the changed order of integration will have:

- \(x\) limits: from \(0\) to \(b\),
- \(y\) limits: depending on \(x\), from

Understand how to change the order of a double integration with the inner limit involving the square root function (b/a)√(a^2-x^2) to b and the outer limit from c to a (where c<a). This problem explores multivariable calculus concepts and applies Fubini's Theorem. Ideal for college-level calculus students.

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