4.6.1 Use substitution to evaluate the following definite integrals. ...
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4.6.2 Use substitution to evaluate the following definite integrals. ...
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4.6.3 Use substitution to evaluate the following definite integrals. ...
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4.6.4 Use substitution to evaluate the following definite integrals. ...
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4.6.5 Use substitution to evaluate the following definite integrals. ...
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4.6.6 Use substitution to evaluate the following definite integrals. ...
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4.6.7 Find
the areas under the following curves. If you use substitution, draw a
graph to compare the original area with that in transformed variables. Area under f (x) = ... from x =0 to x =3.
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4.6.8 Find
the areas under the following curves. If you use substitution, draw a
graph to compare the original area with that in transformed variables. Area under g(x) = ...from x = 0 to x = ln(2).
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4.6.10 Find
the areas under the following curves. If you use substitution, draw a
graph to compare the original area with that in transformed variables. Area under ...from t =0 to t =2.
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4.6.10 Find
the areas under the following curves. If you use substitution, draw a
graph to compare the original area with that in transformed variables. Area under ...from t =0 to t =2.
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4.6.11 Find
the areas under the following curves. If you use substitution, draw a
graph to compare the original area with that in transformed variables. Area under ...from y =0 to y =2
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4.6.12 Find
the areas under the following curves. If you use substitution, draw a
graph to compare the original area with that in transformed variables. Area under s(z) = sin(z + π) from z =0 to z =π.
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4.6.13 The definite integral can be used to find the area between two curves. In each case,
a. Sketch the graphs of the two functions over the given range, and shade the area between the curves.
b.
Sketch the graph of the difference between the two curves. The area
under this curve matches the area between the original curves.
c. Find the area under the difference curve (remembering to use absolute value). ...
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4.6.14 The definite integral can be used to find the area between two curves. In each case,
a. Sketch the graphs of the two functions over the given range, and shade the area between the curves.
b.
Sketch the graph of the difference between the two curves. The area
under this curve matches the area between the original curves.
c. Find the area under the difference curve (remembering to use absolute value). ...
Get solution
4.6.15 The definite integral can be used to find the area between two curves. In each case,
a. Sketch the graphs of the two functions over the given range, and shade the area between the curves.
b.
Sketch the graph of the difference between the two curves. The area
under this curve matches the area between the original curves.
c. Find the area under the difference curve (remembering to use absolute value). ...
Get solution
4.6.16 The definite integral can be used to find the area between two curves. In each case,
a. Sketch the graphs of the two functions over the given range, and shade the area between the curves.
b.
Sketch the graph of the difference between the two curves. The area
under this curve matches the area between the original curves.
c. Find the area under the difference curve (remembering to use absolute value). ...
Get solution
4.6.17 The definite integral can be used to find the area between two curves. In each case,
a. Sketch the graphs of the two functions over the given range, and shade the area between the curves.
b.
Sketch the graph of the difference between the two curves. The area
under this curve matches the area between the original curves.
c. Find the area under the difference curve (remembering to use absolute value). ...
Get solution
4.6.18 The definite integral can be used to find the area between two curves. In each case,
a. Sketch the graphs of the two functions over the given range, and shade the area between the curves.
b.
Sketch the graph of the difference between the two curves. The area
under this curve matches the area between the original curves.
c. Find the area under the difference curve (remembering to use absolute value). f ( x )=sin(2x) and g(x)=cos(2x) for 0≤ x ≤π.
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4.6.19 Find
the average value of the following functions over the given range.
Sketch a graph of the function along with a horizontal line at the
average to make sure that your answer makes sense. ... for 0< x <3.
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4.6.20 Find
the average value of the following functions over the given range.
Sketch a graph of the function along with a horizontal line at the
average to make sure that your answer makes sense. ... for 0.5 < x < 2.0.
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4.6.21 Find
the average value of the following functions over the given range.
Sketch a graph of the function along with a horizontal line at the
average to make sure that your answer makes sense. ...
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4.6.22 Find
the average value of the following functions over the given range.
Sketch a graph of the function along with a horizontal line at the
average to make sure that your answer makes sense. sin(2x) for 0 ≤ x ≤ π/2.
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4.6.23 Use integration by parts to evaluate the following as definite integrals. Find the area under the curve g(x)= x ln(x) for 1≤ x ≤ 2. Sketch a graph to see if your answer makes sense.
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4.6.24 Use integration by parts to evaluate the following as definite integrals. Find the area under the curve g(x)= x sin(2πx) for 0 ≤ x ≤ 2. Sketch a graph to see if your answer makes sense.
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4.6.25 We
have used little vertically oriented rectangles to compute areas. There
is no reason why little horizontal rectangles cannot be used. Here are
the steps to find the area under the curve y = f (x) from x = 0 to x =1 by using such horizontal rectangles. a.
Draw a picture with five horizontal rectangles, each of height 0.2,
approximately filling the region to the right of the curve.
b. Calculate an upper and lower estimate of the length of each rectangle based on the length of the upper and lower boundaries.
c. Add these up to find upper and lower estimates of the area.
d. Think now of a very thin rectangle at height y. How long is the rectangle?
e. Write down a definite integral expression for the area.
f. Evaluate the integral and check that the answer is correct. ...
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4.6.26 We
have used little vertically oriented rectangles to compute areas. There
is no reason why little horizontal rectangles cannot be used. Here are
the steps to find the area under the curve y = f (x) from x = 0 to x =1 by using such horizontal rectangles. a.
Draw a picture with five horizontal rectangles, each of height 0.2,
approximately filling the region to the right of the curve.
b. Calculate an upper and lower estimate of the length of each rectangle based on the length of the upper and lower boundaries.
c. Add these up to find upper and lower estimates of the area.
d. Think now of a very thin rectangle at height y. How long is the rectangle?
e. Write down a definite integral expression for the area.
f. Evaluate the integral and check that the answer is correct. ...
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4.6.27
There is often more than one way to divide up a region to find an
area or volume. Use the fact that the area of a circle of radius r is ...to find the volume of a cone of height 1 that has radius r at a height r . Think of the cone as being built of a stack of little circular disks with some small thickness Δr .
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4.6.28 There is often more than one way to divide up a region to find an area or volume. Break a sphere of radius r into horizontal discs to find the volume. The trick is to figure out the area of each disc at height z where z ranges from Δr to r .
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4.6.29 Some books define the natural log function with the definite integral as the function l(a) for which ... Using this definition, we can prove the laws of logarithms (page 84). Show that l(6) − l(3)=l(2).
(Use the summation property of the definite integral to write the
difference as an integral, and then use the substitution ...
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4.6.30 Some books define the natural log function with the definite integral as the function l(a) for which ... Using this definition, we can prove the laws of logarithms (page 84). Find the integral from a to 2a by following the same steps. (Make the substitution ...
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4.6.31 Some books define the natural log function with the definite integral as the function l(a) for which ... Using this definition, we can prove the laws of logarithms (page 84). ...
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4.6.32 Some books define the natural log function with the definite integral as the function l(a) for which ... Using this definition, we can prove the laws of logarithms (page 84). ...
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4.6.33 The
average of a step function computed with the definite integral matches
the average computed in the usual way. Test this in the following
situations by finding the average of the values directly, and then as
the integral of a step function. Suppose a math
class has four equally weighted tests. A student gets 60 on the first
test, 70 on the second, 80 on the third, and 90 on the last.
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4.6.34 The
average of a step function computed with the definite integral matches
the average computed in the usual way. Test this in the following
situations by finding the average of the values directly, and then as
the integral of a step function. A math class has 20 students.
In a quiz worth 10 points, 4 students get 6, 7 students get 7, 5
students get 8, 3 students get 9, and 1 student gets 10.
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4.6.35 Suppose water is entering a tank at a rate of g(t)= ...where g is measured in liters per hour and t is
measured in hours. The rate is 0 at times 0, 15, and 24. Find the
total amount of water entering during the first 15 hr, from t = 0 to t = 15. Find the average rate at which water entered during this time.
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4.6.36 Suppose water is entering a tank at a rate of g(t)= ...where g is measured in liters per hour and t is measured in hours. The rate is 0 at times 0, 15, and 24. Find the total amount and average rate from t = 15 to t = 24.
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4.6.37 Suppose water is entering a tank at a rate of g(t)= ...where g is measured in liters per hour and t is measured in hours. The rate is 0 at times 0, 15, and 24. Find the total amount and average rate from t = 0 to t = 24.
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4.6.38 Suppose water is entering a tank at a rate of g(t)= ...where g is measured in liters per hour and t is
measured in hours. The rate is 0 at times 0, 15, and 24. Suppose that
energy is produced at a rate of ... in J/h (Joules per hour).
Find the total energy generated from t = 0 to t = 24. Find the average rate of energy production. Check that g(t) changes sign from positive to negative at t =15.
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4.6.39 Several very skinny 2.0 m long snakes are collected in the Amazon. Each has density of ρ(x) given by the following formulas, where ρ is measured in g/cm and x is measured in centimeters from the tip of the tail. For each snake,
a. Find the minimum and maximum density of the snake. Where does the maximum occur?
b. Find the total mass of the snake.
c. Find the average density of the snake. How does this compare with the minimum and maximum?
d. Graph the density and average. ...
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4.6.40 Several very skinny 2.0 m long snakes are collected in the Amazon. Each has density of ρ(x) given by the following formulas, where ρ is measured in g/cm and x is measured in centimeters from the tip of the tail. For each snake,
a. Find the minimum and maximum density of the snake. Where does the maximum occur?
b. Find the total mass of the snake.
c. Find the average density of the snake. How does this compare with the minimum and maximum?
d. Graph the density and average. ...
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4.6.41 A piece of E. coli DNA has about 4.7 ×... nm long. The genetic code consists of four possible nucleotides, called A, C, G, and T. For each of the following cases,
a.
Use the given information to find the formula for the numberof A’s,
C’s, G’s and T’s per thousand as a function of distance along the DNA
strand.
b. Find the total number of A’s, C’s, G’s, and T’s in the DNA.
c. Find the mean number of A’s, C’s, G’s, and T’s in the DNA per thousand. Suppose
that the number of A’s per thousand increases linearly from 150 at one
end of the DNA strand to 300 at the other. The number of C’s per
thousand decreases linearly from 350 at one end to 200 at the other, and
the number of G’s per thousand increases linearly from 220 at one end
to 320 at the other. The remainder is made up of T’s.
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4.6.42 A piece of E. coli DNA has about 4.7 ×... nm long. The genetic code consists of four possible nucleotides, called A, C, G, and T. For each of the following cases,
a.
Use the given information to find the formula for the numberof A’s,
C’s, G’s and T’s per thousand as a function of distance along the DNA
strand.
b. Find the total number of A’s, C’s, G’s, and T’s in the DNA.
c. Find the mean number of A’s, C’s, G’s, and T’s in the DNA per thousand. Suppose
that the number of A’s per thousand increases linearly from 200 at one
end of the DNA strand to 250 at the other. The number of C’s per
thousand increases linearly from 250 at one end to 300 at the other, and
the number of G’s per thousand decreases linearly from 300 at one end
to 200 at the other. The remainder is made up of T’s.
Get solution
4.6.44 Suppose
water is entering a series of vessels at the given rate. In each case,
find the total amount of water entering during the first second, and the
average rate during that time. Compare the average rate with rate at
the “average time,” at t = 0.5 halfway through the time
period from 0 to 1. In which case is the average rate greater than the
rate at the average time? Graph the flow rate function, and mark the
flow rate at the average time. Can you guess what it is about the shape
of the graph that determines how the average rate compares with the rate
at the average time? Water is entering at a rate of ...
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4.6.44 Suppose
water is entering a series of vessels at the given rate. In each case,
find the total amount of water entering during the first second, and the
average rate during that time. Compare the average rate with rate at
the “average time,” at t = 0.5 halfway through the time
period from 0 to 1. In which case is the average rate greater than the
rate at the average time? Graph the flow rate function, and mark the
flow rate at the average time. Can you guess what it is about the shape
of the graph that determines how the average rate compares with the rate
at the average time? Water is entering at a rate of ...
Get solution
4.6.45 Suppose
water is entering a series of vessels at the given rate. In each case,
find the total amount of water entering during the first second, and the
average rate during that time. Compare the average rate with rate at
the “average time,” at t = 0.5 halfway through the time
period from 0 to 1. In which case is the average rate greater than the
rate at the average time? Graph the flow rate function, and mark the
flow rate at the average time. Can you guess what it is about the shape
of the graph that determines how the average rate compares with the rate
at the average time? Water is entering at a rate of ...
Get solution
4.6.46 Suppose
water is entering a series of vessels at the given rate. In each case,
find the total amount of water entering during the first second, and the
average rate during that time. Compare the average rate with rate at
the “average time,” at t = 0.5 halfway through the time
period from 0 to 1. In which case is the average rate greater than the
rate at the average time? Graph the flow rate function, and mark the
flow rate at the average time. Can you guess what it is about the shape
of the graph that determines how the average rate compares with the rate
at the average time? Water is entering at a rate of 4t (1 − t) ...
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4.6.47 Find the area between the two curves f (x)=cos(x) and g(x)=0.1x for 0≤ x ≤10.
a. Graph the two functions and make the three regions.
b. Have your computer find where each region begins and ends.
c. Integrate to find the area of each region.
d. Add them up.
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4.6.48 Suppose the volume of water in a vessel obeys the differential equation ... with V (0) = 0.
a. Graph the functions f and V for t = 0 to t = 2 and label the curves.
b. Find the volume at time t =
10. What is the definite integral that has the same answer? Shade the
area on your graph from part a and write the associated integral.
c. Define a function A(T ) that gives the average rate of change of volume as a function of time (the total volume added between times t = 0 and t = T divided by the elapsed time). Graph this on the same graph as f (t). Label the curves (and write the formula for A(t) as a definite integral). Why is the average rate A greater than the instantaneous rate f ?
d. Graph f (t) between t = 0 and t = 10 and the constant function with a rate equal to the average at time T =
10. What is the area under the line? Does it match what you found in
part b? Why should it? Mark the point where the average and
instantaneous rates are equal. Use your computer to solve for this
point.
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