3. Area Between 2 Curves

by M. Bourne

math image

We are trying to find the area between 2 curves, y1 = f1(x) and y2 = f2(x) and the lines x = a and x = b.

We see that if we subtract the area under lower curve

y1 = f1(x)

from the area under the upper curve

y2 = f2(x),

then we will find the required area. This can be achieved in one step:

math image

Alternative Way to Find The Formula (First Principles)

Another way of deriving this formula is as follows (the thinking here is important for understanding how we develop the later formulas in this section).

Each "typical" rectangle indicated has width Δx and height y2y1, so its area is (y2y1x.

If we add all these typical rectangles, starting from a and finishing at b, the area is approximately:

area between sum

Now if we let Δx → 0, we can find the exact area by integration:

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Likewise, we can sum vertically by re-expressing both functions so that they are functions of y and we find:

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Example:

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Find the area between the curves y = x2 + 5x and y = 3 − x2 between x = -2 and x = 0.

Answer

 

Exercises

1. Find the area bounded by y = x3, x = 0 and y = 3.

 

Answer

 


2. Find the area bounded by the curves

y = x2 + 5x and y = 3 − x2.

(This is an extension of the Example above.)


Here is the answer using LiveMath:

LIVEMath


Normal answer:

Answer

 


3. Find the area bounded by the curves

y = x2, y = 2 − x and y = 1.

Answer


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