Explanation 1) Concept: By using substitution rule for definite integrals and fundamental rule of calculus 2) Theorem: Fundamental theorem of calculus: If f is continuous on [ a , b ] , then ∫ a b f x d x = F b - F a . Substitution rule for definite integrals: If g ' is continuous on [ a , b ] and f is continuous on the range of u = g ( x ) then ∫ a b f g x g ' x d x = ∫ g ( a ) g ( b ) f ( u ) d u 3) Formula: ∫ sin ⁡ x d x = - cos ⁡ x + C 3) Given: ∫ 0 1 sin ⁡ ( 3 π t ) d t 4) Calculation: Consider, ∫ 0 1 sin ⁡ ( 3 π t ) d t Since sin ⁡ ( 3 π t ) is continuous [0,1], the given integral exists. By applying substitution rule for definite integrals, Let u = 3 π t

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Chapter 4.R, Problem 20E

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∫01sin⁡(3πt)dt if it exists

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Chapter 4.R, Problem 19E

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Chapter 4 Solutions

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Ch. 4.1 - Prob. 1E Ch. 4.1 - a Use six rectangles to find estimates of each...Ch. 4.1 - a Estimate the area under the graph of f(x)=1/x...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Evaluate the upper and lower sums for...Ch. 4.1 - With a programmable calculator or a computer, it...Ch. 4.1 - Prob. 10E

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∫ 0 1 sin ⁡ ( 3 π t ) d t if it exists

Solution Summary: The author explains substitution rule for definite integrals and fundamental rule of calculus. If g' is continuous on [a,b], the given integral exists.

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To evaluate:

∫01sin⁡(3πt)dt if it exists

Want to see the full answer?

Chapter 4 Solutions

∫ 0 1 sin ⁡ ( 3 π t ) d t if it exists

Solution Summary: The author explains substitution rule for definite integrals and fundamental rule of calculus. If g' is continuous on [a,b], the given integral exists.

Videos

To evaluate: ∫01sin⁡(3πt)dt if it exists

Want to see the full answer?

Chapter 4 Solutions

To evaluate:

∫01sin⁡(3πt)dt if it exists