For the curve \(y=\sqrt{x}\), between \(x=0\) and \(x=2\), find: The area under the curve.

Understanding the concept of a definite integral is a key part in finding the area under a curve.
A definite integral represents the area under the curve of a given function, between two specified points.
Consider it as the sum of an infinite number of infinitesimally small areas.
When setting up a definite integral, you include limits of integration, which indicate the interval over which you're finding the area.
In our exercise, the function is \( \sqrt{x} \) and the interval is from \( x = 0 \) to \( x = 2 \).

Therefore, the integral is set up as: \[ \int_{0}^{2} \sqrt{x} \, dx \]
This integral will give us the total area under the curve \( \sqrt{x} \) between \( x = 0 \) and \( x = 2 \).

The power rule for integration is very useful when dealing with functions involving exponents.
It states: \[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]
Here, \( n \) is any real number, and \( C \) is the constant of integration.
In definite integrals, the constant \( C \) isn't included because we evaluate the function at specific limits.

For our function \( \sqrt{x} \), we rewrite it as \( x^{1/2} \) to apply the power rule.
Then, we integrate \( x^{1/2} \): \[ \int x^{1/2} \, dx = \frac{x^{1/2+1}}{1/2+1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]
Using the power rule simplifies the integration process and provides us with the antiderivative we need.

Breaking down the integration process step-by-step is crucial to understanding how to find the area under a curve. Let's go through the steps:

• Step 1: Set up the integral: Start by setting up the definite integral with the given limits: \[ \int_{0}^{2} \sqrt{x} \, dx \].


• Step 2: Simplify the integrand: Rewrite \( \sqrt{x} \) as \( x^{1/2} \) for easier integration: \[ \int_{0}^{2} x^{1/2} \, dx \].


• Step 3: Integrate the function: Apply the power rule for integration: \[ \int_{0}^{2} x^{1/2} \, dx = \left. \frac{2}{3} x^{3/2} \right|_{0}^{2} \].


• Step 4: Evaluate the definite integral: Finally, evaluate the antiderivative at the upper and lower limits: \[ \left. \frac{2}{3} x^{3/2} \right|_{0}^{2} = \frac{2}{3} (2)^{3/2} - \frac{2}{3} (0)^{3/2} = \frac{2}{3} (2\sqrt{2}) = \frac{4\sqrt{2}}{3} \].

By following these steps carefully, you can find the area under the curve accurately and efficiently.