Find the area between the $x$-axis and the curve $y=x^{2} - 4$ between $x=-2$ and $x=2$. What is the integral that will give this area? $\displaystyle \int_{ ＿ }^{ ＿ }$

dx. (Online, to specify limits of integration, enter them here. Lower limit:

. Upper limit:

.)

Now evaluate the definite integral:

What's wrong with this answer?

If we graph $y=x^{2} - 4$, we see that it is below the axis between $-2$ and $2$. What is $f(x_i)$ in our Riemann sums?

In order for the Riemann sum to represent the area, we have to take $|f(x_i)|$ instead of just $f(x_i)$. The definite integral for the area becomes $\displaystyle \int_a^b |f(x)| \, dx$. In this example, we can avoid the absolute values by just multiplying everything by $-1$. If we multiply the final answer by $-1$, we get the area

. That works because $x^{2} - 4$ is negative over the entire interval.

How can we handle area for a curve that goes above and below the axis? Suppose we want to know the area bounded by $y=x^{2} - 2 x$ between $x=0$ and $x=3$. Sketch the graph of this function and the region of integration below. There are two regions: one is below the $x$-axis between $x=$

and $x=$

and the other is above the $x$-axis between $x=$

and $x=$

. If we find the area in each of these two regions, we can find the total area by

them together.

**Feedback from applet**

endpoints:

function:

Write down the integral that represents the area below the axis, and evaluate it:

$\displaystyle \int_{ ＿ }^{ ＿ } $

$dx =$

(Online, enter integral limits here. Lower limit:

. Upper limit:

.)

Write down the integral that represents the area above the axis, and evaluate it:

$\displaystyle \int_{ ＿ }^{ ＿ } $

$dx =$

(Online, enter integral limits here. Lower limit:

. Upper limit:

.)

What is the total area bounded by the $x$-axis and $y=x^{2} - 2 x$ between $x=0$ and $x=3$?

Evaluate a definite integral (or sum of definite integrals) that gives the area between the $x$-axis and the curve $y=- x^{2} + 4 x$ between $x=-1$ and $x=5$.
In the interval $x \in (-1, 5)$, the expression $- x^{2} + 4 x$ is

. We need to split the interval into

pieces.

The intervals are

. Enter in order, separated by commas, such as `(-5,2),(2,7)`.

The sign of $- x^{2} + 4 x$ in the intervals is

. Enter either `positive` or `negative` for each interval, in same order as the intervals, separated by commas, such as `positive, negative`.

Therefore, we need to change the sign in which intervals?

. Enter in order, separated by commas, such as `(-5,2),(2,7)`.

The area over or under each interval is

. Enter in order, separated by commas, such `5,3/5`.

The total area is

.