Area Calculations in a Plane Click here for Calculations of Surface Area of Revolution

In a Nut Shell: Calculation of the area under a curve or between curves is a
three step process such as:

1.	Given y1 = f1(x), y2 = f2(x), and values of x, plot the curves in the xy-plane.
2.	Identify the element of area, dA, and show it on the graph. Typically dA = (yu ˗ yL) dx or dA = (xR ˗ xL) dy
3.	Determine the limits of integration a ≤ x ≤ b or c ≤ y ≤ d for the area to be calculated) by setting y1(x) = y2(x) or x1(y) = x2(y) then evaluate the integral: b d I = ∫ (yu ˗ yL) dx or ∫ (xR ˗ xL) dy a c

One curve bounded by the x-axis and one or more vertical lines.

Example 1: I = ∫ e^-x dx where y(x) = e^-x

Let the area be bounded below by the x-axis and on each side by [0,4]

Steps 1 and 2: Draw curve and show the element of area.

Here dA = (y_u ˗ y_L) dx , A = ∫ [y(x) ˗ 0 ] dx or A = ∫ [ e^-x ˗ 0 ] dx

Step 3: Determine limits of integration. In this case the lower limit is x = 0

and the upper limit is x = 4.

Step 3: Evaluate the integral. Note: Area should always be positive value.

4 4

A = ∫ e^-x dx = - e^-x | = - [ e^-4 - 1 ] = 1 - e^-4

0 0

Click here for another example.

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