Question

The area of the region enclosed by the parabolas \( y = x^2 - 5x \) and \( y = 7x - x^2 \) is _________.

Answer 1

Correct Answer: 72

The provided curves are:

\( y = x^2 - 5x \quad \text{and} \quad y = 7x - x^2 \)

Let the functions be:

\( f(x) = x^2 - 5x \quad \text{and} \quad g(x) = 7x - x^2 \)

The area enclosed by these curves is computed as:

\( \int_0^6 (g(x) - f(x)) \, dx = \int_0^6 ((7x - x^2) - (x^2 - 5x)) \, dx \)

The integrand simplifies to:

\( = \int_0^6 (12x - 2x^2) \, dx \)

Integrating term-by-term yields:

\( = \left[ \frac{12x^2}{2} - \frac{2x^3}{3} \right]_0^6 \)

Evaluating at the limits:

\( = (6 \cdot 6^2) - \frac{2}{3} \cdot (6)^3 \)

\( = 216 - 144 = 72 \, \text{unit}^2 \)