Integration

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Signed area function

Lesson

As we've seen, we need to be mindful whether the area we're calculating is above or below the x-axis as this influences how we set up or finalise our calculation.

Relating back to what we saw on the Fundamental Theorem of Calculus, we can write our area function as $F(x)$`F`(`x`). Calculating the integral of $f(x)$`f`(`x`) will give us the expression for $F(x)$`F`(`x`).

But if the area we want is bellow the x-axis, then we'll need to write our area function as $-F(x)$−`F`(`x`).

Let's take a look at some examples.

Determine an expression for the area function that can be used to find the area shaded on the graph.

We begin by finding the equation of the function graphed.

Since the area is above the $x$`x`-axis, our area function will simply be the integral of $f(x)$`f`(`x`).

Note here that adding our constant of integration is optional. Since we'll be using this function to calculate area, when we subtract the lower limit from the upper limit that $c$`c` value will cancel out. So you don't actually need it!

Graph the function $f(x)=(x-1)(x+2)$`f`(`x`)=(`x`−1)(`x`+2) and determine an expression for the area function that can be used to calculate the area bounded by the curve and the $x$`x`-axis between the $x$`x`-intercepts.

First we graph our function and shade the area we're interested in.

Since the area is below the $x$`x`-axis we'll need to introduce a negative into our area function.

Consider the function $f\left(x\right)=8-4x$`f`(`x`)=8−4`x`.

Graph the function.

Loading Graph...What is the $x$

`x`-coordinate of the $x$`x`-intercept?Let $F\left(x\right)$

`F`(`x`) be the area function representing the area bound by $f\left(x\right)$`f`(`x`) and the $x$`x`-axis for $x<2$`x`<2.Integrate $f\left(x\right)$

`f`(`x`) to find an expression for $F\left(x\right)$`F`(`x`).Use $C$

`C`as the constant of integration.

Consider the function $f\left(x\right)=\left(x+2\right)\left(x-4\right)$`f`(`x`)=(`x`+2)(`x`−4).

Graph the function.

Loading Graph...What are the $x$

`x`-coordinates of the $x$`x`-intercepts?Write both values on the same line, separated by a comma.

Let $F\left(x\right)$

`F`(`x`) be the area function representing the area bound by $f\left(x\right)$`f`(`x`) and the $x$`x`-axis for $-2−2< `x`<4.Integrate $f\left(x\right)$

`f`(`x`) to find an expression for $F\left(x\right)$`F`(`x`).Use $C$

`C`as the constant of integration.

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

Apply integration methods in solving problems