Calculus 1 : Area
Study concepts, example questions & explanations for Calculus 1
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Example Questions
Example Question #211 : Regions
Calculate the area of the region between the function and the x-axis on the interval 
The area of the region between a function and the x-axis on the interval 
Because 
In order to solve the integral we use the inverse power rule which says
And because integrating is a linear operation we apply the power rule term-by-term to get
And by the corollary of the Fundamental Theorem of Calculus
Example Question #131 : Area
Find the area of the region bound by the y-axis, h(t), and the lines 
Find the area of the region bound by the y-axis, h(t), and the lines 
First, we want to set up our integral:
Next, recall that 
Now, to find the area, we need to find the H(6)-H(5)
Simplify to get:
So, we can round our answer to get:
Example Question #132 : How To Find Area Of A Region
Find the area between the curves of 
Set this up as a definite integral to find volume. In order to do this, we need to know which function is above the other. In this case, 
To solve for area 
Notice that 
Therefore:
By the fundamental theorem of calculus:
Example Question #3091 : Functions
Calculate the area between 
We can find the interval that we are supposed to integrate over by setting the two equations equal to each other.
Manipulating the equation you get
Factoring this gives you,
Therefore we integrate between 1 and 9.
Next we see that if we were to graph both equations, 
Example Question #212 : Regions
Many times people forget that area under a curve can be found, not only by integration, but also sometimes by using simple geometry.
Find the area under the following function
on the interval 
It is not possible
There are two different ways to solve this problem. You can integrate both pieces of the piecewise function separately and add them together to get the area or you can look at the problem in a more geometric way.
First we graph the equation:
.
We can actually make this into two right triangles
.
The area of a triangle is known as
The red triangle has a height of 
The blue triangle has a height of 
Therefore the area of this function on the interval 
Example Question #135 : How To Find Area Of A Region
Which of the following is an expression that describes the area enclosed by 
The question is invalid.
The answer is not shown.
The first step is to write the first equation as a function of x. You will note however that this is the equation for a circle and is thus not a function of x. This ends up not being a problem however since 
The next step is to find the points in which these two functions intersect by setting them equal to each other.
This yields the points 
Yet since it is symmetric, we can rewrite it as
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