6.5 Interpreting the Behavior of Accumulation Functions Involving Area

Learning objectives

—FUN-5.A—Represent accumulation functions using definite integrals—Topic 6.5• FUN-5.A Represent accumulation functions using definite integrals.• FUN-5.A.3 Graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as g(x) = ∫ₐˣ f(t)dt.• Enduring understanding FUN-5: The Fundamental Theorem of Calculus connects differentiation and integration.

For $g(x)=\int_a^x f(t)\,dt$ , the value $g(x)$ is the signed area accumulated from $a$ to $x$ , while the Fundamental Theorem gives $g'(x)=f(x)$ . Thus a representation of $f$ reveals both how much has accumulated and how $g$ is changing.

Information about $f$	Conclusion about $g$
$f>0$ / $f<0$	$g$ increases / decreases
$f$ changes $+\to-$ / $-\to+$	$g$ has a local maximum / minimum
$f$ increases / decreases	$g$ is concave up / concave down, since $g''=f'$
signed area from $a$ to $x$	value of $g(x)$

Suppose the signed area under $f$ from $a$ to $c$ is $5$ , and the area from $c$ to $d$ lies below the axis with magnitude $2$ . Then $g(c)=5$ and $g(d)=5-2=3$ . On $(c,d)$ , $f<0$ , so $g$ is decreasing, yet $g$ remains positive. If $f$ changes from positive to negative at $c$ , then $g$ has a local maximum there.

A zero of $f$ means $g'(x)=0$ ; it is only a critical point of $g$ , not automatically a zero of $g$ . A zero of $g$ occurs when the net signed area from $a$ to $x$ is zero. Likewise, $f>0$ tells whether $g$ increases, not whether $g$ itself is positive.

Next Topic

—6.6 Applying Properties of Definite IntegralsRead