Explanation Formula used: Integration by parts Formula: ∫ u v d x = u ∫ v d x − ∫ [ d u d x ∫ v d x ] . Calculation: Consider the integral ∫ 0 ∞ x 2 e − 3 x d x . Use the above mentioned formula and compute the indefinite integral value. ∫ x 2 e − 3 x d x = x 2 ∫ e − 3 x d x − ∫ [ d d x ( x 2 ) ∫ e − 3 x d x ] d x = x 2 × e − 3 x − 3 − ∫ ( 2 x × e − 3 x − 3 ) d x = − x 2 e − 3 x 3 + 2 3 ∫ x e − 3 x d x = − x 2 e − 3 x 3 + 2 3 [ x ∫ e − 3 x d x − ∫ [ d d x ( x ) ∫ e − 3 x d x ] d x ] On further simplification the above expression becomes, ∫ x 2 e − 3 x d x = − x 2 e − 3 x 3 + 2 3 [ x × e − 3 x − 3 − ∫ ( 1 × e − 3 x − 3 ) d x ] = − x 2 e − 3 x 3 + 2 3 [ − x e − 3 x 3 + 1 3 ∫ e − 3 x d x ] = − x 2 e − 3 x 3 + 2 3 [ − x e − 3 x 3 + 1 3 ( − e

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Chapter 18.3, Problem 2WE

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Chapter 18.3, Problem 1WE

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Calculus III May I please have some elaborations on Example 2 part a? Thank you.

1. A bicyclist is riding their bike along the Chicago Lakefront Trail. The velocity (in feet per second) of the bicyclist is recorded below. Use (a) Simpson's Rule, and (b) the Trapezoidal Rule to estimate the total distance the bicyclist traveled during the 8-second period. t 0 2 4 6 8 V 10 15 12 10 16 2. Find the midpoint rule approximation for (a) n = 4 +5 x²dx using n subintervals. 1° 2 (b) n = 8 36 32 28 36 32 28 24 24 20 20 16 16 12 8- 4 1 2 3 4 5 6 12 8 4 1 2 3 4 5 6

= 5 37 A 4 8 0.5 06 9

Chapter 18 Solutions

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The integral ∫ 0 ∞ x 2 e − 3 x d x .

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Chapter 18 Solutions