What is the area under the curve y = x ^ 2 from x = 0 to x =1?

<h2>What is the area under the curve $y = x^2$ from $x = 0$ to $x = 1$?</h2>

The choices are:

• 1/2

• 1/3

• 1/4

• 2/3

The correct answer is: 1/3.

Let's detail why this is the correct answer using integral calculus.

To determine the area under the curve given the function \(y = x^2\) from \(x = 0\) to \(x = 1\), we use the definite integral. The definite integral can be represented as:

<pre><code> \[ \text{Area} = \int_{0}^{1} x^2 \, dx \] </code></pre>

To compute this, we need to find the antiderivative of \(x^2\). The antiderivative of \(x^2\) is given by:

<pre><code> \[ \int x^2 \, dx = \frac{x^3}{3} + C \] </code></pre>

Where \(C\) is the constant of integration. For definite integrals, we can ignore the constant term and simply evaluate the antiderivative between the limits 0 and 1.

Thus, we evaluate:

<pre><code> \[ \left[ \frac{x^3}{3} \right]_0^1 = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \] </code></pre>

Therefore, the area under the curve \(y = x^2\) from \(x = 0\) to \(x = 1\) is \(\frac{1}{3}\).

This confirms that the correct choice is 1/3.

Understanding the integral calculus process and applying it correctly helps ensure that we get to the right solution. For further reading on integrals and their applications, you may refer to a reputable calculus textbook or an educational resource such as Khan Academy or Coursera.