Area difference

Calculus Level 3

Two curves, $f(x)$ and $g(x)$ are a cubic and parabola respectively. $f(x)$ is in the shape of the graph $y=x^3$ and touches the $x$ -axis at $x=0$ and $x=4$ . $g(x)$ is in the shape of the graph $y=1.5x^2$ and has its vertex placed on the point $(2,-6)$ . What is the difference in their areas below the x-axis between their two roots in common? Note: $f(x)$ has a repeated root at $x=0$ .

7 5⅓ 37⅓ 7⅓

1 solution

Poca Poca
Jul 6, 2018

From the information about the function, we can easily find out that $f(x)=x^2(x-4)=x^3-4x^2$ and $g(x)=1.5x^2-6x$ . Hence, the areas that we are looking for are $A_f = |\int_0^4{(x^3-4x^2)}dx|=\left| \left[\frac{x^4}{4} - \frac{4x^3}{3}\right]_0^4 \right|=\frac{64}{3}$ and $A_g= |\int_0^4{(1.5x^2-6x)}dx|=\left| \left[\frac{1.5x^3}{3} - \frac{6x^2}{2}\right]_0^4 \right|=16$ . Thus, the difference that we are looking for is: $\frac{64}{3} - 16 = \boxed{5.\overline{3}}$

Area difference

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