Question

Using integration, evaluate the area of the region bounded by the curve \( y = x^2 \), the lines \( y = 1 \) and \( y = 3 \), and the y-axis.

Answer 1

Problem Definition: Determine the area of the region enclosed by the curve y = x², the horizontal lines y = 1 and y = 3, and the y-axis (x = 0).

Express x in terms of y: From y = x², we get x = √y.

Integral Setup: The area is computed by integrating x with respect to y from y = 1 to y = 3: Area = ∫[from 1 to 3] √y dy.

Integration: The indefinite integral of √y (or y^(1/2)) is (2/3)y^(3/2).

Definite Integral Calculation: Evaluate the integral from 1 to 3: Area = [(2/3)y^(3/2)] from 1 to 3 = (2/3)[3^(3/2) - 1^(3/2)].

Simplification: Since 3^(3/2) = 3√3, the area is (2/3)(3√3 - 1).

Result: The area of the specified region is (2/3)(3√3 - 1) square units.

Final Answer: (2/3)(3√3 - 1)