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2. Definite Integrals and Numeric Integration Calculus • Calculus answers two very important questions. • The first, how to find the instantaneous rate of change, we answered with our study of derivatives • The second we are now ready to answer, how to find the area of irregular regions. Sigma Notation • Remember sigma notation from precalculus? • The sum of n terms a , a , a , ..., a is written as 1 2 3 n n S a i i 1 • The i tells you where to start and end summing. Approximating Area • We will now approximate an irregular area bounded by a function, the x-axis between vertical lines x=a and x=b, like the one below by finding the areas of many rectangles and summing them up. Finding area • Break region into subintervals (strips) • These strips resemble rectangles • Sum of all the areas of these “rectangles” will give the total area Rectangular Approximation Method (RAM) • Since the height of the rectangle varies along the subinterval, in order to find area of the rectangle, we must use either the left hand endpoint (LRAM) to find the height, the right hand endpoint (RRAM) or the midpoint (MRAM) • The more rectangles you make, the better the approximation Rectangular Approximation Method (RAM) • If a function is increasing, LRAM will underestimate the area and RRAM will overestimate it. • If a function is decreasing, LRAM will overestimate the area and RRAM will underestimate it Trapezoid Approximation • Another approximation we can use (and probably the best) is trapezoids. • Trapezoids give an answer between the LRAM and RRAM • The formula for the area of a trapezoid is ½(x)(y1+y2) Example 1 • A particle starts at x=0 and moves along the x-axis with velocity v(t)=t2. Where is the particle at t=3? Use interval width of ¼ and MRAM. • Find height at each midpoint of interval. Multiply height times width to get area. Sum all the areas. Subinterval [0, ¼] [¼, ½] [½,3/4] [3/4, 1] Midpoint 1/8 3/8 5/8 7/8 Height (t2) 1/64 9/64 25/64 49/64 Area 1/256 9/256 25/256 49/256 • Add all 1/256+9/256+25/256+49/256 +81/256+121/256+169/256+ 225/256+289/256+361/256+ 441/256+529/256 = 8.98 etc Example 2 • Find the area under y=x2 from x=0 to x=3, use width of ½. Use all methods • LRAM 0 (1 / 2) (1 / 2) (1 / 2) 1 (1 / 2) (3 / 2) (1 / 2) 2 (1 / 2) (5 / 2) (1 / 2) 2 2 2 2 2 2 1 / 2[0 (1 / 2) 1 (3 / 2) 2 (5 / 2) ] 2 2 2 2 2 2 6 .8 7 5 • MRAM 1 / 2[(1 / 4) (3 / 4) (5 / 4) (7 / 4) (9 / 4) (11 / 4) ] 2 • RRAM 2 2 2 2 2 1 / 2[(1 / 2) (1) (3 / 2) (2) (5 / 2) (3) ] 2 2 2 2 2 2 8 .9 3 7 5 1 1 .3 7 5 • Trap 1 / 2(1 / 2)[(0 2(1 / 2) 2(1) 2(3 / 2) 2(2) 2(5 / 2) (3) ] 2 2 2 2 2 2 2 9.125 Example 3 • Find the area under y=x2+2x-3 from x=0 to x=2, use width of ½ • LRAM 1 / 2[ 3 1.75 0 2.25] 1 .2 5 • MRAM 1 / 2[ 2.4375 .9375 1.0625 3.5625] • RRAM 1 / 2[ 1.75 0 2.25 5] .6 2 5 2 .7 5 • Trap (1 / 2)(1 / 2)[ 3 2( 1.75) 2(0) 2(2.25) 5] 0.75 How many rectangles should we make? • The estimate of area gets more and more accurate as the number of rectangles (n) gets larger How many rectangles should we make? • If we take the limit as n approaches infinity, we should get the exact area • We will take more about this tomorrow….. Worksheet notation L n L R A M w ith n intervals M n M R A M w ith n intervals R n R R A M w ith n intervals T n T rapezoids w ith n intervals If interval is 3 units long and you have 6 subintervals, Each subinterval will be 3/6 or ½ wide. BREAK!! Remember from yesterday….. • We were talking about increasing the number of rectangles giving us a better estimate of the area • What if we took the limit as n approached infinity?? • The area approximation would approach the actual area • The process of finding the sum of areas of rectangles to approximate area of a region is called a Riemann Sum, after Bernhard Riemann Riemann Sums • Riemann proved that the finite process of adding up the rectangular areas could be found by a process known as definite integration. Here is the essence of his great, time-saving work. b b A lim n ia f ( xi ) x a f ( x ) dx What the notation means Upper limit of integration b Integral symbol Integrand f ( x ) dx a Variable of Integration Lower limit of integration Read this as integral from a to b of f of x dx Example 4 4 • Evaluate 2 xdx geometrically as well as 0 on your calculator Negative area? • The example we just looked at was nonnegative on the interval we evaluated. This is not always the case. • If f(x) is non-negative and integratable over a closed interval [a,b] then the area under the curve is the definite integral of f from a to b • If f(x) is negative and integrable over a closed interval [a,b], then the area under the curve is the OPPOSITE of the definite integral of f from a to b. b A f ( x ) dx a In general… b • f ( x ) dx does NOT give us area but rather the NET accumulation over the interval x=a to x=b. If f(x) is positive and negative on a closed interval, then f ( x ) dx will NOT give us area. a b a • When integrating left to right, regions above the x-axis are positive and regions below the x-axis are negative. • When integrating right to left, regions above the x-axis are negative and regions below the x-axis are positive. • This can be summarized as b a a f ( x ) dx f ( x ) dx b Negative functions • When using definite integrals to find area, you must divide the interval into subintervals where function is positive and where it is negative and use absolute values of definite integral • When using area to find definite integrals, you must assign the correct sign to the area. Example 5 • The graph of f(x) is shown below. If A1 and A2 are positive numbers that represent the areas of the shaded regions, then find the following. Another property of definite integrals • The property that allows us to do the calculations in the previous example is b a c f ( x ) dx a b f ( x ) dx c f ( x ) dx Example 6 • Approximate ( x 4) dx using four subintervals of equal length and trapezoidal method. Test your answer against the calculator’s approximation using fnint. Can any of these approximations represent the area of the region? Why or why not? 5 2 1 Example 7 • Find the area in the previous problem using trapezoids and also set up integrals needed to calculate with calculator. Another way to find area with calculator b A a f ( x ) dx Example 8 • We can also find areas when our function is given to us in either data form or graph form. 3 f ( x ) dx • Approximate using LRAM, MRAM, RRAM, and trapezoids. Do these represent area? Also approximate f’(1) 0 Example 9 8 Approximate f ( x ) dx using LRAM, RRAM, and trapezoids. Do these represent area? Why did I leave off MRAM? Also approximate f’(7) 1 Example 10 • Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula. Decide if the integral represents the area. Example 11 0 5 f ( x ) dx f ( x ) dx • Evaluate and . Do these represent the area of the region? Why or why not? If not, what is the area of the region? 7 2 Properties of definite integrals • We have seen some of these already. Example 12 • Given that 1 4 f ( x ) dx 5, 1 1 f ( x ) dx 2, h ( x ) dx 7 1 1 • Find 1 1 f ( x ) dx 2 f ( x )dx can 't d o h ( x )dx can 't d o 4 4 0 2 f ( x ) dx 3 1 2 1 2 f ( x ) 3 h ( x ) dx 1 31 Example 13 8 10 • If 0 f ( x ) dx 17 and 0 f ( x ) dx 12 10 , find (3 f ( x ) 2) dx 8 Example 14 9 x2 x3 5 6 • dx using area of the region to help evaluate the integral. • If sin xdx 2 Example 15 use this fact and symmetry of the graph of sin x to find the following. 0