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April 11th, 2017, 07:05 PM   #1
Joined: Feb 2017
From: henderson

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Unhappy Use the following graph to find the values of the integrals.

Enter the exact answers.

Help guys! I do not know what to do! My tutor told me to use rectangle, triangle and circle formulas!
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April 12th, 2017, 02:01 AM   #2
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Each integral can be evaluated as a combination of areas, adding areas above the x-axis and subtracting areas below the x-axis. That's not a full specification, but you should have been taught this method.

The first integral corresponds to a square (of side length 2) and two congruent triangles that "cancel" (as one is above the x-axis and the other is below the x-axis. Hence its value is 4.

The second integral corresponds to half of a circle with radius 4. Hence its value is 8$\pi$ (as the entire area is above the x-axis).

Is that sufficient help?
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April 12th, 2017, 04:50 AM   #3
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From x= 0 to x= 2, the region under the graph (down to y= 0) is a 1 by 1 rectangle. I

From x= 2 to 3 the region under the graph is a triangle with base 1 height 2.

Now the graph goes below the x-axis so will have negative "signed area". The region from 3 to 4 above the graph (up to y= 0) is a triangle with base 1 and height 2.

The region from 4 to 6 is a triangle with height and base 2 but is still below the x-axis so its signed area is negative

The region from x= 6 to x= 14 is a semi-circle with radius 5 above the x-axis.

The region from x= 14 to x= 20 is again below the x-axis and is a trapezoid with bases of length 6 and 3 and height 4.

The integral "from a to b" is the sum of the areas (including the negatives for regions below the x-axis) of the regions between x= a and x= b.
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