Question

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

• A. \( \frac{13}{3} \) sq. units
• B. \( \frac{10}{3} \) sq. units
• C. \( \frac{16}{3} \) sq. units
• D. \( \frac{19}{3} \) sq. units

Hint:

To find the area between a curve and the x-axis, integrate the function over the given interval and evaluate the definite integral.

Answer 1

The Correct Option is B

Step 1: Set up the integral for the area.
The area between the curve \( y = x^2 + 1 \) and the x-axis from \( x = 1 \) to \( x = 2 \) is given by the integral: \[ A = \int_1^2 (x^2 + 1) \, dx \]
Step 2: Integrate the function.
We can compute the integral: \[ A = \int_1^2 x^2 \, dx + \int_1^2 1 \, dx \] This gives: \[ A = \left[ \frac{x^3}{3} \right]_1^2 + \left[ x \right]_1^2 \] Evaluating the limits: \[ A = \left( \frac{2^3}{3} - \frac{1^3}{3} \right) + (2 - 1) \] \[ A = \frac{8}{3} - \frac{1}{3} + 1 = \frac{7}{3} + 1 = \frac{10}{3} \]
Step 3: Conclusion.
The area is \( \frac{10}{3} \) sq. units.