Find the area bounded by two curves Y=√x and Y=x^3

Answer 1

To find the area bounded by the two curves \(y = \sqrt{x}\) and \(y = x^3\), we need to set up an integral.

First, we need to find the points of intersection between the two curves:

\(\sqrt{x} = x^3\)

Squaring both sides to eliminate the square root:

\(x = x^6\)

Rearranging the equation:

\(x^6 - x = 0\)

Factoring out an x:

\(x(x^5 - 1) = 0\)

So the solutions are \(x = 0\) and \(x = 1\).

Now, we need to set up the integral to find the area between the two curves:

\(\text{Area} = \int_{0}^{1} (x^3 - \sqrt{x}) dx\)

Evaluating this integral will give you the area bounded by the two curves.