Approximate Area using Sums of Areas

Example: Let f(x) be a continuous function that satisfies the following conditions:

14

∫ f(x) dx = 25

6

and 14

∫ f(x) dx = 15

20

20

Find: ∫ [ f(x) + 2 ] dx

6

20 20 20

Note: ∫ [ f(x) + 2 ] dx = ∫ f(x) dx + ∫ 2 dx The second integral yields 28.

6 6 6

14 20

Note: ∫ f(x) dx = ˗ ∫ f(x) dx = ˗ 15

20 14

20 14 20

∫ f(x) dx = ∫ f(x) dx ˗ ∫ f(x) dx = 25 ˗ 15 = 10

6 6 14

20

∫ [ f(x) + 2 ] dx = 10 + 28 = 38 (result)

6

2

Find: ∫ 48 x² f(2x³ + 4) dx

1

Strategy: Use a change in variables. Let w = 2 x³ + 4 So dw = 6 x² dx

20 20

which gives ∫ 8 f(w) dw And from the first part: ∫ f(x) dx = 10

6 6

20

So ∫ 8 f(w) dw = 8 (10) = 80 (result)

6

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