Question

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

Answer 1

Given curves:

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

Let

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

To find the area enclosed between these curves, we calculate:

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

Simplify the integrand:

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

Now, integrate term by term:

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

Substitute the limits:

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

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