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MAT1102_calc_l20_wk13_2solutions
Course: MAT 1102, Fall 2009School: Allan Hancock College
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MAT1102 Algebra & Calculus I Review question Calculus Which integration techniques would you use? 1. 2. 3. 4. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 1 Outline Additional Integration Techniques Integration using Tables Approximate Integration Areas Between Curves Readings for next week MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part...
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MAT1102 Algebra & Calculus I Review question Calculus Which integration techniques would you use? 1. 2. 3. 4. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 1 Outline Additional Integration Techniques Integration using Tables Approximate Integration Areas Between Curves Readings for next week MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 2 Smart Trig Substitution Proof that the area of a circle is r2. More writing space on next slide MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 3 Smart Trig Substitution More writing space MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 4 Another example Find Use the substitution MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 5 Integration Using Tables is the wave function used for electron behaviour. Find Number of the list of standard integrals (back of textbook) is: MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 6 Integration Using Tables Find Number of the list of standard integrals (back of textbook) is: Complete the square on x2 +4x+5 to make it look like a2 +u2. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 7 Integration Using Tables Then substitute if necessary to simplify. Then use the standard integral. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 8 Approximate Integration Left Endpoint <a href="/keyword/riemann-sums/" ><a href="/keyword/riemann-sum/" >riemann sum</a> s</a> h=0.1; x=0:h:2-h; y=exp(-x.^2); left=sum(y*h) Right Endpoint <a href="/keyword/riemann-sums/" ><a href="/keyword/riemann-sum/" >riemann sum</a> s</a> h=0.1; x=h:h:2; y=exp(-x.^2); right=sum(y*h) Midpoint <a href="/keyword/riemann-sums/" ><a href="/keyword/riemann-sum/" >riemann sum</a> s</a> h=0.1; x=h/2:h:2-h/2; y=exp(-x.^2); mid=sum(y*h) Trapezoidal rule average of left and right sums, trap = (left+right)/2 MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 9 Example Find estimates for the value of Note that this is one of those integrals we cannot find by standard means. Substitution won't work. First plot the curve. The value correct to 4 decimal places is 0.8821. For h = 0.1, see which of left, right, mid and trapezoidal estimates is best. The differences are: exact-[left right mid trap]=0.8821-[0.9311 0.8329 0.8821 0.8820] =[-0.0490 0.0491 -0.0000 0.0001] So, the mid point rule is accurate to 4 decimal places, the trapezoidal rule is also pretty good. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 10 Areas Between Curves How do we find the area between two curves defined by y = f(x) and y = g(x)? Area = , so long as f is above g over the domain [a, b]...
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