Question

Find the area enclosed between the x-axis and the curve, if y = x^2 − 25 for (-5) is less than or equal to x less than or equal to 5.

a. 50

b. (-500)

c. 500

d. 25

Answer 1

Find the area enclosed between the x-axis and the curve, if $y = x^2 - 25$ for (-5) is less than or equal to x less than or equal to 5.

a. 50

b. (-500)

c. 500

d. 25

The area A is given by the integral from $x = -5$ to $x = 5$ of the curve $y = x^2 - 5$:

$$\begin{array}{l} A = \int_{-5}^{5} (x^2 - 25) \, dx = \left(\frac{x^3}{3} - 25x\right) \big|_{-5}^{5} \\ = \left(\left(\frac{5^3}{3} - 25 * 5\right) - \left(\frac{(-5)^3}{3} - 25 * (-5)\right)\right) = -2 * 125 * \frac{2}{3} \\ = -\frac{500}{3} \end{array}$$

and area A is below the x-axis; and, as we see, the sign of the value A is negative. The actual value of the area is $+\frac{500}{3}$

Answer: $\frac{500}{3}$