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Problem 5.R.105a
Area functions and the Fundamental Theorem Consider the function
ƒ(t) = { t if ―2 ≤ t < 0
t²/2 if 0 ≤ t ≤ 2
and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.
(a) Evaluate F(―2) and F(2).
Problem 5.R.86
Evaluating integrals Evaluate the following integrals.
∫₀⁵ |2𝓍―8|d𝓍
Problem 5.R.9d
Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
Problem 5.R.96c
Velocity to displacement An object travels on the 𝓍-axis with a velocity given by v(t) = 2t + 5, for 0 ≤ t ≤ 4.
(c) True or false: The object would travel as far as in part (a) if it traveled at its average velocity (a constant), for 0 ≤ t ≤ 4. .
Problem 5.R.1a
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.
(a) A(𝓍) = ∫ₐˣ ƒ(t) dt and ƒ(t) = 2t―3 , then A is a quadratic function.
Problem 5.R.43
Evaluating integrals Evaluate the following integrals.
∫₀¹ √𝓍 (√𝓍 + 1) d𝓍
Problem 5.R.65
Evaluating integrals Evaluate the following integrals.
∫₀^²π cos² 𝓍/6 d𝓍
Problem 5.R.30
Evaluate the following derivatives.
d/d𝓍 ∫₃ᵉˣ cos t² dt
Problem 5.R.64
Evaluating integrals Evaluate the following integrals.
∫ y² (3y³ + 1)⁴ dy
Problem 5.R.57
Evaluating integrals Evaluate the following integrals.
∫₀² (2𝓍 + 1)³ d𝓍
Problem 5.R.11
Limit 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.
∫₀² (𝓍²―4) d𝓍
Problem 5.R.26
Use geometry and properties of integrals to evaluate the following definite integrals.
∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 . (Hint: Write the integral as sum of two integrals.)
Problem 5.R.105b
Area functions and the Fundamental Theorem Consider the function
ƒ(t) = { t if ―2 ≤ t < 0
t²/2 if 0 ≤ t ≤ 2
and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.
(b) Use the Fundamental Theorem to find an expression for F '(𝓍) for ―2 ≤ 𝓍 < 0.
Problem 5.R.5
Estimate ∫₁⁴ √(4𝓍 + 1) d𝓍 by evaluating the left, right, and midpoint Riemann sums using a regular partition with n = 6 subintervals.
Problem 5.R.107
Limits with integrals Evaluate the following limits.
lim ∫₂ˣ eᵗ² dt
𝓍→2 ---------------
𝓍 ― 2
Problem 5.R.39
Evaluating integrals Evaluate the following integrals.
∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍
Problem 5.RE.15e
Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.
(e) ∫₋₂² 3𝓍ƒ(𝓍)d𝓍
Problem 5.RE.15c
Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.
(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍
Problem 5.RE.15a
Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.
(a) ∫₋₄⁴ ƒ(𝓍) d𝓍
Problem 5.3.14a
Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.
(a) A(2)
Problem 5.5.95a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.
(a) ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍 = ½ (ƒ(𝓍))² + C.
Problem 5.3.18a
Area functions for the same linear function Let ƒ(t) = 2t ― 2 and consider the two area functions A (𝓍) = ∫₁ˣ ƒ(t) dt and F(𝓍) = ∫₄ˣ ƒ(t) dt .
(a) Evaluate A (2) and A (3). Then use geometry to find an expression for A (𝓍) , for 𝓍 ≥ 1 .
Problem 5.1.71a
Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(a) Describe the motion of the object over the interval [0,6].
Problem 5.2.77a
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
∫₀⁴ (4𝓍― 𝓍²) d𝓍
Problem 5.2.51a
Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(a) ∫₄⁰ 3𝓍(4 ― 𝓍) d(𝓍)
Problem 5.5.100a
Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.
(a) ∫¹₋₁ 𝓍ƒ(𝓍²) d𝓍
Problem 5.1.5a
The velocity in ft/s of an object moving along a line is given by v = ƒ(t) on the interval 0 ≤ t ≤ 6 (see figure), where t is measured in seconds.
(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)
Problem 5.3.87a
Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.
Problem 5.1.59a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) Consider the linear function ƒ(𝓍) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.
Problem 5.3.20a
Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).
(a) Find and graph the area function A (𝓍) = ∫ₐˣ ƒ(t) dt .
ƒ(t) = 2t + 5 , a = 0
