text: # Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that
  
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  1 Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these
  
  2 Mth 43 Section 6 Are Under the Grph of Nonnegtive Function If y f is nonnegtive nd integrle over the intervl,, then the re under the curve y f over, is given y f d If the curve is sometimes negtive, then one cn split the region into pieces using the roots of the function s the limits of the integrl Consider the function whose grph is given elow: Theorem: If f is continuous on, nd if c For the function shown ove, Are of f ( d ) nd c Are of f ( d ) f( d ) c c c f d f d f d, then
  
  3 Mth 43 Section 6 Emple : Given the grph of f, if the re of is nd re of is 8, find c f d Emple 6: Given f d, f d, the curve nd the -is from = - to = 4 4 f d 4 Find the re etween
  
  4 Mth 43 Section 6 Other Properties of Definite Integrl Assume tht f nd g re continuous functions, where k is constnt numer kd k If f g over,, then f d g d 3 If m f M over,, then m f d M f d f d 4 If f is n odd function, then f d If f is n even function, then f d f d 3
  
  5 Mth 43 Section 6 Try this one: Find the lower sum for f () =, [, ] if the prtition is 3 P,,,, 4 4 Try this one: Estimte 6 3 d y using left endpoint estimtes, where n = 6 Try this one: 3 6 Given f( ) d 3, f( ) d, f( ) d 9, find f( ) d 3 6 4
  
  6 Mth 43 Section 6 Section 6: The Fundmentl Theorem of Clculus Recll from the previous section tht: For function f which is continuous on,, there is one nd only one numer tht stisfies the inequlity f, for ll prtitions P of L P I U P f, This unique numer I is clled the definite integrl (or just the integrl) of f from to nd is denoted y f ( d ) This numer cn e positive, negtive or zero Let f e continuous function over the intervl upper limit vries etween nd, Define new function y F f t dt Here, the F cn e seen s the re under the grph of f from F s the ccumulted re function If f hppens to e nonnegtive function, then to We cn think of So for emple, F f t dt if if then this is re under the function f from = to = F f t dt then this is re under the function f from = to =
  
  7 Mth 43 Section 6 Emple : If f is the function whose grph is given elow, nd F f t dt, find F Fundmentl Theorem of Clculus Prt If f is continuous function over the intervl is continuous on, nd differentile on F f t dt, Note: If = then =,, then the function F f t dt d d, Moreover, F ' f t dt f, for ll in Emple : Let f e continuous function stisfying 3 f t dt Stte F() Find c Find F' f( ) F'' f '( ) Hence, where f is positive, F is incresing where f is negtive, F is decresing where f is zero, F hs possile m, min or inflection point where f is incresing, F is concve up where f is decresing, F is concve down
  
  8 Mth 43 Section 6 Emple 3: Find '( ) F ( t t) dt F given Emple 4: Find d cos ( s) ds d Then find '(4 ) F d Emple : Find 3t dt d Recll the following definite integrl property from Section 6: f () d f() d
  
  9 Mth 43 Section 6 Other times if the upper limit is not just, we ll need to use the chin rule Tht is, d v( ) f ( tdt ) f( v ( )) v'( ) d Emple 6: Find '( ) F given F dt t And yet, other times, if oth limits of integrtion re functions of we ll pply the following rule: d d v( ) u( ) f ( tdt ) f( v ( )) v'( ) f( u ( )) u'( ) Emple 7: Given 3 t dt, find F F' Let f e continuous function over the intervl, A function F is clled n ntiderivtive for f over the intervl, if F is continuous on, nd F' f for ll in, Emple 8: Give few ntiderivtives for f
  
  10 Mth 43 Section 6 Theorem: Fundmentl Theorem of Clculus Prt Let f e continuous function over the intervl, If G is ny ntiderivvite for f over the intervl,, then f d GG Emple 9: Clculte the definite integrl d using FTOC
  
  11 Mth 43 Section 6 F ( t 4 t ) dt Try this one: Let the function F e defined y 3 Find ny criticl numers for F d d Recll: F ' f t dt f Discuss the concvity of F
  
  ### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
  
  Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
  
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  1 Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these
  
  2 Mth 43 Section 6 Are Under the Grph of Nonnegtive Function If y f is nonnegtive nd integrle over the intervl,, then the re under the curve y f over, is given y f d If the curve is sometimes negtive, then one cn split the region into pieces using the roots of the function s the limits of the integrl Consider the function whose grph is given elow: Theorem: If f is continuous on, nd if c For the function shown ove, Are of f ( d ) nd c Are of f ( d ) f( d ) c c c f d f d f d, then
  
  3 Mth 43 Section 6 Emple : Given the grph of f, if the re of is nd re of is 8, find c f d Emple 6: Given f d, f d, the curve nd the -is from = - to = 4 4 f d 4 Find the re etween
  
  4 Mth 43 Section 6 Other Properties of Definite Integrl Assume tht f nd g re continuous functions, where k is constnt numer kd k If f g over,, then f d g d 3 If m f M over,, then m f d M f d f d 4 If f is n odd function, then f d If f is n even function, then f d f d 3
  
  5 Mth 43 Section 6 Try this one: Find the lower sum for f () =, [, ] if the prtition is 3 P,,,, 4 4 Try this one: Estimte 6 3 d y using left endpoint estimtes, where n = 6 Try this one: 3 6 Given f( ) d 3, f( ) d, f( ) d 9, find f( ) d 3 6 4
  
  6 Mth 43 Section 6 Section 6: The Fundmentl Theorem of Clculus Recll from the previous section tht: For function f which is continuous on,, there is one nd only one numer tht stisfies the inequlity f, for ll prtitions P of L P I U P f, This unique numer I is clled the definite integrl (or just the integrl) of f from to nd is denoted y f ( d ) This numer cn e positive, negtive or zero Let f e continuous function over the intervl upper limit vries etween nd, Define new function y F f t dt Here, the F cn e seen s the re under the grph of f from F s the ccumulted re function If f hppens to e nonnegtive function, then to We cn think of So for emple, F f t dt if if then this is re under the function f from = to = F f t dt then this is re under the function f from = to =
  
  7 Mth 43 Section 6 Emple : If f is the function whose grph is given elow, nd F f t dt, find F Fundmentl Theorem of Clculus Prt If f is continuous function over the intervl is continuous on, nd differentile on F f t dt, Note: If = then =,, then the function F f t dt d d, Moreover, F ' f t dt f, for ll in Emple : Let f e continuous function stisfying 3 f t dt Stte F() Find c Find F' f( ) F'' f '( ) Hence, where f is positive, F is incresing where f is negtive, F is decresing where f is zero, F hs possile m, min or inflection point where f is incresing, F is concve up where f is decresing, F is concve down
  
  8 Mth 43 Section 6 Emple 3: Find '( ) F ( t t) dt F given Emple 4: Find d cos ( s) ds d Then find '(4 ) F d Emple : Find 3t dt d Recll the following definite integrl property from Section 6: f () d f() d
  
  9 Mth 43 Section 6 Other times if the upper limit is not just, we ll need to use the chin rule Tht is, d v( ) f ( tdt ) f( v ( )) v'( ) d Emple 6: Find '( ) F given F dt t And yet, other times, if oth limits of integrtion re functions of we ll pply the following rule: d d v( ) u( ) f ( tdt ) f( v ( )) v'( ) f( u ( )) u'( ) Emple 7: Given 3 t dt, find F F' Let f e continuous function over the intervl, A function F is clled n ntiderivtive for f over the intervl, if F is continuous on, nd F' f for ll in, Emple 8: Give few ntiderivtives for f
  
  10 Mth 43 Section 6 Theorem: Fundmentl Theorem of Clculus Prt Let f e continuous function over the intervl, If G is ny ntiderivvite for f over the intervl,, then f d GG Emple 9: Clculte the definite integrl d using FTOC
  
  11 Mth 43 Section 6 F ( t 4 t ) dt Try this one: Let the function F e defined y 3 Find ny criticl numers for F d d Recll: F ' f t dt f Discuss the concvity of F
  
  ### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
  
  Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
  
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  1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
  
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  ### Section 6.1 Definite Integral
  
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  ### MAT137 Calculus! Lecture 27
  
  MAT37 Clculus! Lecture 7 Tody: More out Integrls (Rest of the Videos) Antiderivtives Next: Fundmentl Theorem of Clculus NEW office hours: T & R @ BA 4 officil wesite http://uoft.me/mat37 Betriz Nvrro-Lmed
  
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  AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
  
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  ### Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
  
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distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: ### The document contains math problems from the Calculus 1 course, specifically focusing on integration and the Fundamental Theorem of Calculus.
  
  Label: multiple-question-and-answer
  
  The document presents multiple questions and answers related to Calculus 1, with a focus on integration and the Fundamental Theorem of Calculus. Here's a brief overview:
  
  1. It starts with practice problems on definite integrals and the Fundamental Theorem of Calculus.
  
  2. There are questions on evaluating definite integrals using the Fundamental Theorem.
  
  3. Several problems involve finding areas under curves and between curves using definite integrals.
  
  4. The document covers properties of definite integrals, including the additivity property.
  
  5. There are examples of using Riemann sums to approximate definite integrals.
  
  6. Questions on the First and Second Fundamental Theorems of Calculus are included.
  
  7. The document also has problems on antiderivatives and indefinite integrals.
  
  8. Some applications of integration are presented, like finding the area between curves.
  
  9. There are practice problems on the substitution rule for definite integrals.
  
  10. The document concludes with a few questions on the relationship between derivatives and integrals.
  
  This document serves as a comprehensive set of practice problems and examples for students learning about