The area of the region bounded by the curve y = x
2
, the x-axis, and the lines x = 1 and x = 2 is:

To find the area of the region bounded by the curve \( y = x^2 \), the x-axis, and the vertical lines \( x = 1 \) and \( x = 2 \), we need to compute the definite integral of the curve between these bounds.

The area \( A \) can be found using the formula:
\[
A = \int_{1}^{2} x^2 \, dx
\]

Let's calculate that step by step:

1. First, find the antiderivative of \( x^2 \):
   \[
   \int x^2 \, dx = \frac{x^3}{3}
   \]

2. Now, evaluate this antiderivative at the limits \( x = 2 \) and \( x = 1 \):
   \[
   A = \left[ \frac{x^3}{3} \right]_{1}^{2}
   \]

3. Substitute the limits:
   \[
   A = \frac{2^3}{3} - \frac{1^3}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}
   \]

Thus, the area of the region is \( \frac{7}{3} \) square units.

Would you like more details on any step?

Here are 5 related questions to deepen your understanding:
1. How would the area change if the bounds were from \( x = 0 \) to \( x = 2 \)?
2. Can you calculate the area under a different curve, like \( y = x^3 \), using the same process?
3. What is the geometric interpretation of a definite integral?
4. How does the Fundamental Theorem of Calculus relate to this problem?
5. How would the process change if the region was bounded by \( y = x^2 \) and \( y = 4x \)?

**Tip:** When calculating areas under curves, always check if the function dips below the x-axis, as you may need to account for negative areas!

The area of the region bounded by the curve y = x^2, the x-axis, and the lines x = 1 and x = 2 is:

Learn how to find the area of the region bounded by the curve y = x^2, the x-axis, and the lines x = 1 and x = 2 using definite integrals. This calculus problem is explained step-by-step, covering key concepts such as the Fundamental Theorem of Calculus and the calculation of definite integrals. Suitable for students in Grades 11-12.

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