Area Under A Curve 

To find the area under a curve that is above the xaxis, 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.
video by rootmath 

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 

video by Michel vanBiezen 

close solution

Problem Statement 

\( y=\sin x ~~ [0,\pi] \)
Solution 

video by Michel vanBiezen 

close solution

Problem Statement 

\(y=x^22x+8 ~~ [1,2]\)
Solution 

video by Michel vanBiezen 

close solution

Problem Statement 

\( y=\sin x \cos x \) \( [\pi/4, \pi/2] \)
Solution 

video by Michel vanBiezen 

close solution

Problem Statement 

\(\displaystyle{ y = \frac{x^2}{\sqrt{x^3+9}} ~~ [1,1] }\)
Solution 

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 xaxis, 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 xaxis for positive area or below the xaxis for negative area.
It is also possible to get zero for the area. This occurs when the area above the xaxis has exactly the same area as the area below the xaxis.
Here is a good video explaining in more detail why the area is negative. He uses an example and explains this very well.
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 

video by Michel vanBiezen 

Final Answer 

0 
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Problem Statement 

Calculate the area under the curve \(y=x^26x\) between 1 and 3.
Final Answer 

Problem Statement 

Calculate the area under the curve \(y=x^26x\) between 1 and 3.
Solution 

Final Answer 

\(46/3\) 
close solution

You CAN Ace Calculus
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. 
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