Answer to Question #345145 in Calculus for Jane

Question

4) Find each of the two areas bounded by the curves y=x³-4x and y = x² + 21.

Answer 1

Find each of the two areas bounded by the curves $y=x^3-4x$ and$y=x^2+2x.$

$$x^3-4x=x^2+2x$$

$$x(x^2-x-6)=0$$

$$x(x+2)(x-3)=0$$

$$x_1=-2, x_2=0, x_3=3$$

$$A_1=\displaystyle\int_{-2}^{0}(x^3-4x-(x^2+2x))dx$$

$$=[\dfrac{x^4}{4}-\dfrac{x^3}{3}-3x^2]\begin{matrix} 0 \\ -2 \end{matrix}$$

$$=0-(\dfrac{(-2)^4}{4}-\dfrac{(-2)^3}{3}-3(-2)^2)$$

$$=\dfrac{16}{3}({units}^2)$$

$$A_2=\displaystyle\int_{0}^{3}(x^2+2x-(x^3-4x))dx$$

$$=[-\dfrac{x^4}{4}+\dfrac{x^3}{3}+3x^2]\begin{matrix} 3 \\ 0 \end{matrix}$$

$$=-\dfrac{(3)^4}{4}+\dfrac{(3)^3}{3}+3(3)^2-0$$

$$=\dfrac{63}{4}({units}^2)$$