Question

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

Short answer

The double integral \(\int_{0}^{1} \int_{x}^{1} \sqrt{1-x^{2}} d y d x\) evaluates to \(\frac{π}{8}\).

Step-by-step solution

Understand the iterated integral ordering

The given double integral is an iterated integral which is a nested series of single integrals. The outer integral is with respect to 'x' and inner one is with respect to 'y'. So, it's going to be easiest to integrate with respect to 'y' first.

Evaluate the inner integral

The inner integral is \(\int_{x}^{1} \sqrt{1-x^{2}} d y\). Note that there are no 'y' variable in the function to integrate. That means the inner integral simply evaluates to \(\sqrt{1-x^{2}}(1-x)\) as y ranges from 'x' to 1.

Evaluate the outer integral

Now consider the outer integral which needs to be evaluated for \(\int_{0}^{1} \sqrt{1-x^{2}}(1-x) dx\). This is a standard single-variable definite integral which can be evaluated using substitution method taking \(u=1-x^{2}\), \(\frac{du}{dx}=-2x\) or power reduction identities, leading to the evaluation of integral as \(\frac{π}{8}\).

Interpret the result

The evaluated integral value of \(\frac{π}{8}\) represents the signed volume under the surface represented by \(f(x, y) = \sqrt{1-x^{2}}\) and above the region in the xy-plane bounded by x from 0 to 1 and y from 'x' to 1 successively.