`The Definite Integral as Area - ClassworkInstead of using the expression &quot;the area under the curve f ( x) between x = a and x = b, we will now denote a shorthand to represent the same thing. We will use what is called &quot;a definite integral.&quot; The definite integral sign is the same as the indefinite integral sign ( ! ) but will contain two limits of integration. The form is as follows:b, f ( x) dx . While this does not seems to make much sense, there is a reason for it. The f ( x) represents the heightaof any one rectangle while the dx represents the width of any one rectangle. So f ( x) dx means the area of any one rectangle. The integral represents the sum of an infinite number of these rectangles. The a represents the starting place for these rectangles while the b represents the ending place for these integrals.f (x )adx f ( x ) dxbab, f ( x ) dxabThe area of one rectangle = f ( x) dxThe sum of an infinite number of rectangle areas. Each rectangle is infinitely thin.When a &lt; b, we are determining the area under the curve from left to right. In that case, our dx is a positivebnumber. If f ( x) is above the axis, then, f ( x) dx will be a positive number.a bWhen b &lt; a, we are determining the area under the curve from right to left. In that case, our dx is a negative number. If f ( x) is above the axis, then, f ( x) dx will be a negative number. This can be summarized below:af ( x) &gt; 0 (curve above axis)f ( x) &lt; 0 (curve below axis)bdx &gt; 0 (left to right) (a &lt; b) dx &lt; 0 (right to left) (b &lt; a)b,a b af ( x) dx &gt; 0 (Area positive)(Area negative), f ( x) dx &lt; 0a b(Area negative) (Area positive), f ( x) dx &lt; 0, f ( x) dx &gt; 0aFurthermore, there are three more rules which will make sense to you:a1. 2. 3., f ( x) dxa b a- If we start at a and end at a, there is no area.,a baf ( x) dx = &quot;c, f ( x) dxbc a- From a to b gives an area. From b to a gives the negative of this area. - Total the area from a to b, add area from b to c = the area from a to c.- 143 Stu Schwartz, f ( x ) dx + , f ( x ) dx = , f ( x ) dxbwww.MasterMathMentor.com AB SolutionsExample) Below you are given the graph of f ( x) formed by lines and a semi-circle. Find the definite integrals.This is the graph off ( x)4131. 4. 7. 10., f (t ) dt = 04 32. 5. 8., f (t ) dt = 20 6 3 103. 6. 9. 12. 15. 18. 21.8, f (t ) dt = 21 3, f (t ) dt = 40 6, f (t ) dt = &quot;1.5 , f (t ) dt = 26 0 10 &quot;3, f (t ) dt = 1.56 6 10, f (t ) dt = 2.50 10, f (t ) dt = &quot;20 &quot;1 &quot;3, f (t ) dt = 2.5 + 20 0 &quot;3 011. &quot; 14. 17. 20.5 5, f (t ) dt = &quot;2.5 &quot; 2, f (t ) dt = &quot;11 , f (t ) dt = 13 + 210 &quot;4, f (t ) dt = 3 , f (t ) dt = &quot;.5, f (t ) dt = 2.5 + 2013. &quot; 16. 19., f (t ) dt = 11 , f (t ) dt = 10.5 , f (t ) dt = 5.5 + 20 5 &quot;20 10&quot;4 10&quot;4 10&quot;4&quot;4, f (t ) dt = 13 + 2h( x) dx = &quot;11, and10,2 f ( t ) dt = &quot;34 &quot; 4 -Suppose5,f ( x ) dx = 18 ,&quot;2,g ( x) dx = 5,&quot;2 5,&quot;2, f ( x) dx = 0, find24.&quot;2 522.&quot;2 5, ( f ( x ) + g ( x )) dx = 23, (g( x ) + 2) dx = 19323.&quot;2 5, [ f ( x ) + g ( x ) &quot; h( x )] dx = 34, ( f ( x ) &quot; 6) dx = &quot;248, 4 g( x ) dx = &quot;20 , h( x &quot; 2) dx = &quot;110 725.26.27.&quot;2&quot;228.&quot;4, g( x + 2) dx = 529., f ( x ) dx = &quot;18530., [ f ( x &quot; 3) + 3] dx = 3918www.MasterMathMentor.com AB Solutions- 144 -Stu SchwartzThe Definite Integral as Area - Homeworkf ( x)1441. 4. 7. 10. 13. 16. 19. 22.,0 5f ( x ) dx = 2 f ( x ) dx = 0 f ( x ) dx = 002. 5. 8. 11. 14. 17. 20. 23.2,2 5f ( x ) dx = 8 f ( x ) dx = 2 f ( x ) dx = 1403. 6. 9. 12. 15. 18. 21. 24.,1 6f ( x ) dx = 12 f ( x ) dx = &quot;2 f ( x ) dx = &quot;40,5 6,4 6,5 2,4,0,35 &quot;3, , ,6f ( x ) dx = &quot;16 f ( x ) dx = 6 f ( x ) dx =94,6 2 &quot;3 &quot;6f ( x ) dx = &quot;14 f ( x ) dx = 0 f ( x ) dx =&quot;940 &quot;3, , ,6 1&quot;3 &quot;3, ,4 0f ( x ) dx = &quot;6 f ( x ) dx = &quot;8 f ( x ) dx =9&quot;6 4&quot;6&quot;3 &quot;6&quot;6 1,&quot;6 1, ,9f ( x ) dx = +8 49f ( x ) dx = &quot; &quot; 8 4 - f ( x ) dx = 64&quot;2,f ( x ) dx = 2f ( x ) dx = 62&quot;2&quot;2,&quot;6, f ( x ) dx = 94- + 2416Suppose that4, f ( x) dx = 2, , f ( x) dx = &quot;1, , f ( x) dx = 7,0 1 2 4evaluate the following: 27.25.,1 1f ( x ) dx = 6 f ( x + 1) dx = &quot;13 326.,0 2 03 f ( x ) dx = 27, f ( x ) dx = 30 428.,029., ( f ( x ) + 3) dx = 830., f ( x &quot; 2) dx = 2231. If, f ( x) dx = &quot;1, find , f ( x) dx0 &quot;3if f is a) even -2 b) odd 0- 145 Stu Schwartzwww.MasterMathMentor.com AB SolutionsThe problems on this page and the next should be done without calculators. 32. For each function below, use a sketch of f to evaluate the 4 definite integrals on top given the statement on the left. (a is a positive constant)2 4 1, f ( x) dx0, f ( x) dx1, f ( x) dx5&quot;9, f ( x ) dx9f ( x) = 3 f ( x) = a f ( x) = &quot; a6 2a &quot;2 a9 3a &quot;3 a--12 &quot;4 a 4 a54 18 a 18 a33. Evaluate the following by making a sketch of the function. 3 3 &quot;1 25 1 25 a. ( x + 2) dx = + = 12 b. x + 2 dx = + = 13 2 2 2 2 &quot;3 &quot;3,,3c.&quot;3,. 91 x + 2 dx = 2 6 + = 21 0 23 / 23d.&quot;3,. 11 2 &quot; x dx = 2 2 + = 5 0 23 / 234. Evaluate1 + x, , f ( x ) dx where f ( x ) = 62 &quot; x , 5027if 0 4 x 4 1 = 2 . Make a sketch of the function. if 1 &lt; x 4 2www.MasterMathMentor.com AB Solutions- 146 -Stu Schwartz`

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