17Calculus Integrals - Area Under Curves
Area Under A Curve |
|---|
To find the area under a curve that is above the x-axis, you just need to integrate the curve between two specific points. This is a natural idea from definite integrals. Here are a couple of videos explaining this.
rootmath - Intro to Area Under a Curve [4min-28secs]
video by rootmath |
|---|
rootmath - Exact Area Under A Curve [9min-55secs]
video by rootmath |
|---|
Okay, to really understand this idea, let's work these practice problems.
Unless otherwise instructed, calculate the area under these curves, between the two points, if given.
Problem Statement |
|---|
\( y=1/x^2 ~~ [1,4] \)
Solution |
|---|
2117 video
video by Michel vanBiezen |
|---|
|
close solution
|
Problem Statement |
|---|
\( y=\sin x ~~ [0,\pi] \)
Solution |
|---|
2118 video
video by Michel vanBiezen |
|---|
|
close solution
|
Problem Statement |
|---|
\(y=x^2-2x+8 ~~ [1,2]\)
Solution |
|---|
2116 video
video by Michel vanBiezen |
|---|
|
close solution
|
Problem Statement |
|---|
\( y=\sin x \cos x \) \( [\pi/4, \pi/2] \)
Solution |
|---|
2119 video
video by Michel vanBiezen |
|---|
|
close solution
|
Problem Statement |
|---|
\(\displaystyle{ y = \frac{x^2}{\sqrt{x^3+9}} ~~ [-1,1] }\)
Solution |
|---|
2120 video
video by Michel vanBiezen |
|---|
|
close solution
|
When Area Under The Curve is Zero or Negative |
|---|
Logically, area should always be positive. However, there will be times when you work the problem correctly and you end up with a negative number. What's the deal?
That's easy. When the area is below the x-axis, the area will turn out to be negative. So it is not that we have negative area. The sign tells us where MOST of the area is, above the x-axis for positive area or below the x-axis for negative area.
It is also possible to get zero for the area. This occurs when the area above the x-axis has exactly the same area as the area below the x-axis.
Here is a good video explaining in more detail why the area is negative. He uses an example and explains this very well.
Section 5.3 - Integrals and Negative Area [5min-21secs]
From this discussion, you now understand why knowing what the graphs look like is so important. Let's work these practice problems. Some of these will have negative or zero area. Make sure you understand why.
Problem Statement |
|---|
Calculate the area under the curve \( y = 60x - 6x^2 \) on \( 0 \leq x \leq 15 \).
Final Answer |
|---|
Problem Statement |
|---|
Calculate the area under the curve \( y = 60x - 6x^2 \) on \( 0 \leq x \leq 15 \).
Solution |
|---|
2433 video
video by Michel vanBiezen |
|---|
Final Answer |
|---|
0 |
|
close solution
|
Problem Statement |
|---|
Calculate the area under the curve \(y=x^2-6x\) between 1 and 3.
Final Answer |
|---|
Problem Statement |
|---|
Calculate the area under the curve \(y=x^2-6x\) between 1 and 3.
Solution |
|---|
2434 video
Final Answer |
|---|
\(-46/3\) |
|
close solution
|
You CAN Ace Calculus
Topics You Need To Understand For This Page
Related Topics and Links
related topics on other pages |
|---|
For discussion of area under a parametric curve, see the parametric calculus page. |
For discussion of area under a polar curve, see the polar calculus page. |
external links you may find helpful |
To bookmark this page and practice problems, log in to your account or set up a free account.
Calculus Topics Listed Alphabetically
Single Variable Calculus |
|---|
Multi-Variable Calculus |
|---|
Differential Equations Topics Listed Alphabetically
Precalculus Topics Listed Alphabetically
Search Practice Problems
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
|
| |
|
Help Keep 17Calculus Free |
|---|
The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free. |
