Finding the Area with Integration

Finding the area of space from the curve of a function to an axis on the Cartesian
plane is a fundamental component in calculus. Definite integration finds the accumulation
of quantities, which has become a basic tool in calculus and has numerous applications
in science and engineering. While it is used to make formulas in physics more comprehensible,
often it is used to optimize the use of space in a given area.

Definite Integration

Whenever we are calculating area in a given interval, we are using definite integration.
Lets try to find the area under a function for a given interval.

(1) Integrate

from [-2, 2].

Step 1: Set up the integral.

Step 2: Find the Integral.

*Note: We don’t have to add a “+C” at the end because it will cancel out finding
the area anyway.

Step 3: Integrate from the given interval, [-2,2].

The area of the curve to the x axis from -2 to 2 is 323 units squared.

On the graph, the red below the parabola is the area and the dotted line is the
integral function. Notice that the integral function is cubic and the original function
is quadratic. The integral will always be a degree higher than the original function.
Looking at the graph, there is a geometric relationship between the original function
and the integral function. We can see at x = -2 the integral function has a y value
of a little under -5, and at x = 2 the integral has a y value of a little over 5.
The difference of 5.3 and -5.3 gives us an area of 323, which is a little over
10.

When taking the definite integral over an interval, sometimes we will get negative
area because the graph interprets area above the x axis as positive area and below
the x axis as negative area.

Find the Area with Integration Examples

(2) Let’s take the integral of y = x from [-3, 1].

We end up with an area of -4.

Looking at the graph at x = -3 and 1, the integral function has an F(x) of 4.5 and
0.5 respectively. Subtracting the lower bound value (4.5) from the upper bound value
(0.5) will yield -4. If we wanted to find the total area, we could take the absolute
value of each bound and add them together to get 5.

It is possible to integrate a function that is not continuous, but sometimes we
need to break up the area into two different integrals.

(3) Here is the function.

Before integrating, we should graph this function to see what it looks like.

Evaluate each integrand.

a)

b)

a) Since the original function is not continuous, we need to look at the
bounds first to see if we are integrating through any discontinuous points. Since
[3,10] is greater than 1, it is continuous and we can integrate using one integral.

b) This integral is a little different. The interval is discontinuous from
[-2,2], so we need to split it into two integrals and add them together.

We have one integral with the interval [-2,1] and the other from [1,2]. The first
interval is less than 1 and the second is greater than 1.

After we integrate, we plug in the specific bounds for both.

The area from [-2,2] is 14 units squared and the area from [3,10] is 35 units squared.

(4) Find the general integral for the yellow shaded region

The area is the integral of f minus the area of g.

(5) Find the area of the purple region bounded by three lines:

First, we need to find the three points of intersection to establish our intervals
for integration. We set each function equal and solve for x.

Like (4), we have to subtract integrals, but we have two seperate quantities to
add.

We have done many examples of integrating to find the area of a curve and it’s relationship
to the integral function. We can also use integration to find the volume
of a 3D object in space, mainly by rotation around an axis.

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