(a) Explanation Given information: Consider the definite integral ∫ − 3 3 ( 2 x + 1 ) d x . Formula used: Let f is continuous function defined on an interval [ a , b ] , n is a positive integer. Δ x = b − a n , x k = a + k Δ x . The right sum of n- rectangles is = ∑ k = 1 n f ( x k ) Δ x Calculation: From the given data, we have [ a , b ] = [ − 3 , 3 ] Δ x = b − a n = 3 − ( − 3 ) n = 6 n x k = a + k Δ x = − 3 + k ( 6 / n ) = − 3 + 6 k n The right sum is given by (b) To determine To find: The value of definite integral with the given values of n. (c) To determine To find: The exact value of the given definite integral as n tends to infinite.

14th Edition

ISBN: 9781464151088

Author: Associate Professor Laura Taalman, Peter Kohn

Publisher: W.H. Freeman & Company

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Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

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Chapter 4.3, Problem 42S

(a)

To determine

To find: The formula for n- rectangle right sum of the given definite integral f on [a,b].

(b)

To determine

To find: The value of definite integral with the given values of n.

(c)

To determine

To find: The exact value of the given definite integral as n tends to infinite.

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Chapter 4.3, Problem 41S

Chapter 4.3, Problem 43S

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Taylor Series Approximation Example- H.W More terms used implies better approximation f(x) 4 f(x) Zero order f(x + 1) = f(x;) First order f(x; + 1) = f(x;) + f'(x;)h 1.0 Second order 0.5 True f(x + 1) = f(x) + f'(x)h + ƒ"(x;) h2 2! f(x+1) 0 x; = 0 x+1 = 1 x h f(x)=0.1x4-0.15x³- 0.5x2 -0.25x + 1.2 51 Taylor Series Approximation H.w: Smaller step size implies smaller error Errors f(x) + f(x,) Zero order f(x,+ 1) = f(x) First order 1.0 0.5 Reduced step size Second order True f(x + 1) = f(x) + f'(x)h f(x; + 1) = f(x) + f'(x)h + "(xi) h2 f(x,+1) O x₁ = 0 x+1=1 Using Taylor Series Expansion estimate f(1.35) with x0 =0.75 with 5 iterations (or & s= 5%) for f(x)=0.1x 0.15x³-0.5x²- 0.25x + 1.2 52

Could you explain this using the formula I attached and polar coorindates

Could you explain this using the formula I attached and polar coordinates

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The formula for n- rectangle right sum of the given definite integral f on [ a , b ] .

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To find: The formula for n- rectangle right sum of the given definite integral f on [a,b].

To find: The value of definite integral with the given values of n.

To find: The exact value of the given definite integral as n tends to infinite.

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Chapter 4 Solutions