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

area under curves youtube playlist

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

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.

more calculus help

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.

\(y=x^2 ~~ [0,4]\)

Problem Statement

\(y=x^2 ~~ [0,4]\)

Solution

2114 solution video

video by Michel vanBiezen

close solution

\(y=x^3 ~~ [1,3]\)

Problem Statement

\(y=x^3 ~~ [1,3]\)

Solution

2115 solution video

video by Michel vanBiezen

close solution

\( y=1/x^2 ~~ [1,4] \)

Problem Statement

\( y=1/x^2 ~~ [1,4] \)

Solution

2117 solution video

video by Michel vanBiezen

close solution

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

Problem Statement

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

Solution

2118 solution video

video by Michel vanBiezen

close solution

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

Problem Statement

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

Solution

2116 solution video

video by Michel vanBiezen

close solution

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

Problem Statement

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

Solution

2119 solution video

video by Michel vanBiezen

close solution

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

Problem Statement

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

Solution

2120 solution 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.

Calculate the area under the curve \( y = 60x - 6x^2 \) on \( 0 \leq x \leq 15 \).

Problem Statement

Calculate the area under the curve \( y = 60x - 6x^2 \) on \( 0 \leq x \leq 15 \).

Final Answer

0

Problem Statement

Calculate the area under the curve \( y = 60x - 6x^2 \) on \( 0 \leq x \leq 15 \).

Solution

2433 solution video

video by Michel vanBiezen

Final Answer

0

close solution

Calculate the area under the curve \(y=x^2-6x\) between 1 and 3.

Problem Statement

Calculate the area under the curve \(y=x^2-6x\) between 1 and 3.

Final Answer

\(-46/3\)

Problem Statement

Calculate the area under the curve \(y=x^2-6x\) between 1 and 3.

Solution

2434 solution video

Final Answer

\(-46/3\)

close solution
Real Time Web Analytics