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

Strategy Revisited

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

Example: Find the area between the intersecting curves y₁(x) = x and y₂ (x) = ½ x²

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

dA = (y_top - y_bottom) dx

Step 3: Determine the limits of integration by finding the points of intersection of the

curves y₁(x) and y₂ (x). To do so set y₁(x) = y₂ (x) so x = ½ x²

and x(1 – 0.5 x) = 0 or x = 0 and x = 2 are the points of intersection.

Step 3: Evaluate the integral:

2 2

A = ∫[x – ½ x²] dx = [x² – x³ / 6] = 2 – 4/3 = 2/3

0 0

Click here to evaluate area using a horizontal rectangle.

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