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April 11th, 2017, 07:05 PM  #1 
Member Joined: Feb 2017 From: henderson Posts: 36 Thanks: 0  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! HOW? 
April 12th, 2017, 02:01 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,939 Thanks: 2210 
Each integral can be evaluated as a combination of areas, adding areas above the xaxis and subtracting areas below the xaxis. 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 xaxis and the other is below the xaxis. 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 xaxis). Is that sufficient help? 
April 12th, 2017, 04:50 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
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 xaxis 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 xaxis so its signed area is negative The region from x= 6 to x= 14 is a semicircle with radius 5 above the xaxis. The region from x= 14 to x= 20 is again below the xaxis 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 xaxis) of the regions between x= a and x= b. 

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