# Without finding an antiderivative of f(x), calculate ʃ(superscript 1)(subscript 0) f(x) dx. How would a graph of an antiderivative of f(x) look? The 2 graphs of a function f can be found at the...

Without finding an antiderivative of f(x),

calculate ʃ(superscript 1)(subscript 0) f(x) dx.

How would a graph of an antiderivative of f(x) look?

The 2 graphs of a function f can be found at the following links:

http://s19.postimage.org/d17tg2khf/calc_homework_24_question_9_graph_1.png

Without finding an antiderivative of f(x), calculate ʃ(superscript 1)(subscript 0) f(x) dx.

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### 1 Answer

You should evaluate the areas under the given curve, between x = 0 and x = 1. Notice that you should evaluate the areas of three triangles, hence, evaluating the total area yields:

`A = A_1 + A_2 + A_3`

Notice that the areas of the first and the second triangles, are the areas of isosceles triangles.

`A_1 = (b_1*h_1)/2`

`A_1 = (1/6*1/6)/2 =gt A_1 = 1/72`

`A_2 = (b_2*h_2)/2 =gt A_2 = (1/2 - 1/6)*(1/3)/2`

`A_2 = (1/3*1/3)/2 =gt A_2 = 1/18`

`A_3 = (b_3*h_3)/2 =gt A_3 = 1*1/2 =gt A_3 = 1/2`

`A = 1/72 + 1/18 + 1/2`

`A = (1 + 8 + 36)/72 =gt A = 45/72 =gt A = 15/26`

**Hence, evaluating the total area found below of the given curve, under the given conditions, yields `A = 15/26` .**