Explanation Given: The integral is ∫ 0 2 x e − x 2 d x and n = 4 Approach: Trapezoidal rule- Let f be a continuous function on interval [ a , b ] . We subdivide the interval into n number of pieces. Length of each subinterval is Δ x = b − a n End points of each subinterval is x 0 = a x 1 = a + Δ x x 2 = a + 2 Δ x x 3 = a + 3 Δ x . . . x n = a + n Δ x Hence approximation of integral according to Trapezoidal rule is ∫ a b f ( x ) d x = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + 2 f ( x 3 ) + 2 f ( x 4 ) + ........... + 2 f ( x n − 1 ) + f ( x n ) ] Calculation: Given integral is ∫ 0 2 x e − x 2 d x We have a = 0 , b = 2 Therefore length of each subinterval is Δ x = 2 − 0 4 = 1 2 = 0.5 End point of each subinterval is x 0 = 0 x 1 = 0 + 0.5 = 0.5 x 2 = 0 + 2 × 0.5 = 1 x 3 = 0 + 3 × 0.5 = 1.5 x 4 = 0 + 4 × 0.5 = 2 We have to find value of function at each end point of interval. Hence value of function at x = 0 f ( 0 ) = 0 × e − 0 = 0 Value of function at x = 0.5 f ( 0.5 ) = 0.5 × e − ( 0.5 ) 2 = 0.3894004 Value of function at x = 1 f ( 1 ) = 1 × e − ( 1 ) 2 = 0.367879 Value of function at x = 1.5 f ( 1.5 ) = 1.5 × e − ( 1

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Chapter 8.CR, Problem 18CR

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Approximate the integral ∫02xe−x2dx using Trapezoidal rule with n=4 and also compare it with exact value of integral.

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Ch. 8.1 - YOUR TURN Use the trapezoidal rule with n=4 to...Ch. 8.1 - Prob. 2YT Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E

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Approximate the integral ∫ 0 2 x e − x 2 d x using Trapezoidal rule with n = 4 and also compare it with exact value of integral.

Solution Summary: The author explains how to approximate the integral using Trapezoidal rule with n=4 and compare it with exact value of integral.

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To find:

Approximate the integral ∫02xe−x2dx using Trapezoidal rule with n=4 and also compare it with exact value of integral.

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Chapter 8 Solutions

Approximate the integral ∫ 0 2 x e − x 2 d x using Trapezoidal rule with n = 4 and also compare it with exact value of integral.

Solution Summary: The author explains how to approximate the integral using Trapezoidal rule with n=4 and compare it with exact value of integral.

Concept explainers

Videos

To find: Approximate the integral ∫02xe−x2dx using Trapezoidal rule with n=4 and also compare it with exact value of integral.

Want to see the full answer?

Chapter 8 Solutions

To find:

Approximate the integral ∫02xe−x2dx using Trapezoidal rule with n=4 and also compare it with exact value of integral.