BackDefinite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.
∫₀⁴ ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5 if 𝓍 ≤ 2
3𝓍 ― 1 if 𝓍 > 2
Evaluating integrals Evaluate the following integrals.
∫₋₅⁵ ω³ /√(ω⁵⁰ + ω²⁰ + 1) dω (Hint: Use symmetry . )
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If ƒ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.
Evaluating integrals Evaluate the following integrals.
∫₋π/₂^π/² (cos 2𝓍 + cos 𝓍 sin 𝓍 ― 3 sin 𝓍⁵) d𝓍
7–64. Integration review Evaluate the following integrals.
60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx
If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.
a. ∫²₋₂ ƒ(x) dx
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫²₋₂ (x² + x³) dx
Given the following definite integral of the function , write the simplified integral:
7–64. Integration review Evaluate the following integrals.
12. ∫ from -5 to 0 of dx / √(4 - x)
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫²⁰⁰₋₂₀₀ 2x⁵ dx
A slowing race Starting at the same time and place, Abe and Bob race, running at velocities u(t) = 4 / (t + 1) mi/hr and v(t) = 4e^(−t/2) mi/hr, respectively, for t ≥ 0.
b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(f) ∫ₐᵇ (2 ƒ(𝓍) ―3g (𝓍)) d𝓍 = 2 ∫ₐᵇ ƒ(𝓍) d𝓍 + 3 ∫₆ᵃ g(𝓍) d𝓍 .
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫π/₁₆^π/⁸ 8 csc² 4𝓍 d𝓍
If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.
e. ∫⁵₋₂ ( ƒ(x) + g(x) ) dx
5
108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].