Explanation Given: y = ln x , [ 1 , 3 ] & n=6 Key concept applied: Arc Length: We drive a formula for the length of a curve y = f ( x ) on an interval [ a , b ] . We will assume that f is continuous and differentiable on an interval [ a , b ] and we will assume that its derivative f ' is also continues on the interval [ a , b ] . We use Riemann sums to approximate the length of the curve over the interval and then take the limit to get an interval. L = ∫ a b 1 + ( d y d x ) 2 d x Midpoint: If we use the endpoints of the subintervals to approximate the integral, we run the risk that the values at the endpoints do not accurately represent the average value of the subinterval. A point which is much more likely to be close to the average would be the midpoint of each subinterval. Using the midpoint in the sum is called the midpoint rule. On the i -interval | x i − 1 , x i | we will call the midpoint x i ¯ that is Δ x [ ∑ i = 1 n f ( x i − 1 + x i 2 ) ] According to Midpoint rule: ∫ a b f ( x ) d x ≈ Δ x [ ∑ i = 1 n f ( x i − 1 + x i 2 ) ] w h e r e Δ x = ( b − a ) n Here b = Upper limit a = Lower limit n = No of interval Formula Used: ∫ a b f ( x ) d x ≈ Δ x [ ∑ i = 1 n f ( x i − 1 + x i 2 ) ] w h e r e Δ x = ( b − a ) n L = ∫ a b 1 + ( d y d x ) 2 d x Calculation: Description: y = ln x , [ 1 , 3 ] & n=6 Step 1: Find the value of Δ x Δ x = 3 − 1 6 = 2 6 = 1 3 a = 1 , b = 3 a n d n = 6 Step 2: Divide interval [ 1 , 3 ] into n=6 subinterval of length Δ x = 1 3 with the following endpoints...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…

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Chapter 9.2, Problem 15E

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The arc length of the curve over the interval using the Midpoint rule

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2. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.4.006.MI. Use the Table of Integrals to evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) 7y2 y² 11 dy Need Help? Read It Master It SUBMIT ANSWER 3. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.4.009. Use the Table of Integrals to evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) tan³(12/z) dz Need Help? Read It Watch It SUBMIT ANSWER 4. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.4.014. Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.) 5 sinб12x dx Need Help? Read It

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y"-9y+20y= 80t-156 y(0) = −6, y'(0) = 5 y(t) =

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The arc length of the curve over the interval using the Midpoint rule

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To Find: The arc length of the curve over the interval using the Midpoint rule

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Chapter 9 Solutions

To Find:

The arc length of the curve over the interval using the Midpoint rule