Question

Find the area of the region bounded by the curve \( y^2 = 4x \), the X-axis, and the lines \( x = 1 \), \( x = 4 \) for \( y \geq 0 \).

Hint:

For area under a parabola, integrate the curve function over the given x-limits.

Answer 1

The curve \( y^2 = 4x \Rightarrow y = \sqrt{4x} = 2\sqrt{x} \) (since \( y \geq 0 \)). Area between \( x = 1 \), \( x = 4 \), X-axis, and curve: \[ \text{Area} = \int_1^4 y \, dx = \int_1^4 2 \sqrt{x} \, dx = 2 \int_1^4 x^{1/2} \, dx. \] \[ \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}. \] \[ 2 \cdot \frac{2}{3} \left[ x^{3/2} \right]_1^4 = \frac{4}{3} \left[ 4^{3/2} - 1^{3/2} \right] = \frac{4}{3} [8 - 1] = \frac{4}{3} \cdot 7 = \frac{28}{3}. \] Answer: \( \frac{28}{3} \) square units.