QuestionFind the total area of the shaded regions.
The area is $\square$ . (Simplify your answer.)

Studdy Solution
Evaluate the integral:
$$A = \int_{-3}^{3} (4x - x^4) \, dx$$
Calculate the antiderivative:
$$\int (4x - x^4) \, dx = 2x^2 - \frac{x^5}{5} + C$$
Evaluate from $-3$ to $3$:
$$A = \left[ 2x^2 - \frac{x^5}{5} \right]_{-3}^{3}$$
Calculate the definite integral:
$$A = \left( 2(3)^2 - \frac{(3)^5}{5} \right) - \left( 2(-3)^2 - \frac{(-3)^5}{5} \right)$$
$$A = \left( 18 - \frac{243}{5} \right) - \left( 18 + \frac{243}{5} \right)$$
$$A = \left( 18 - 48.6 \right) - \left( 18 + 48.6 \right)$$
$$A = -30.6 - 66.6$$
$$A = -97.2$$
Since area cannot be negative, take the absolute value:
$$A = 97.2$$
The total area of the shaded regions is:
$$\boxed{97.2}$$