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05:50Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer.
β«βΒ² (πΒ²β4) dπ
Use geometry and properties of integrals to evaluate the following definite integrals.
β«ββ° (2π + β(16βπΒ²)) dπ . (Hint: Write the integral as sum of two integrals.)
Area functions and the Fundamental Theorem Consider the function
Ζ(t) = { t if β2 β€ t < 0
tΒ²/2 if 0 β€ t β€ 2
and its graph shown below. Let F(π) = β«ββΛ£ Ζ(t) dt and G(π) = β«ββΛ£ Ζ(t) dt.
(b) Use the Fundamental Theorem to find an expression for F '(π) for β2 β€ π < 0.
Evaluating integrals Evaluate the following integrals.
β« yΒ² /(yΒ³ + 27) dy
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ and Ζ' are continuous functions for all real numbers.
(f) β«βα΅ (2 Ζ(π) β3g (π)) dπ = 2 β«βα΅ Ζ(π) dπ + 3 β«βα΅ g(π) dπ .
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ and Ζ' are continuous functions for all real numbers.
(b) Given an area function A(π) = β«βΛ£ Ζ(t) dt and an antiderivative F of Ζ, it follows that A'(π) = F(π) .
