The rate of U.S. per capita sales of bottled water for the period 2007–2014 could be approximated by s(t) = 0.25t2 − t + 29 gallons per year    (0 ≤ t ≤ 7), where t is time in years since the start of 2007.† Use a definite integral to estimate the total U.S. per capita sales of bottled water from the start of 2009 to the start of 2011. (Round your answer to the nearest gallon.)

Q: Find the indefinite integral. (Use C for the constant of integration.) 9 - ex dx e7x

A: Given integration is : ∫9-exe7xdx  Use the formula of integration, ∫eaxdx=eaxa+C

Q: The rate of new tree growth (in millions of cubic meters per year) in the Sudan t years after 1980…

A:

Q: Find the indefinite integral by making a change of variables.

A: Given: ∫(x+1)2-xdx

Q: Find the indefinite integral. (Use C for the constant of integration.) 8 – ex e3x

A: Given that: ∫8-exe3x.dx

Q: Find the indefinite integral. (Use C for the constant of integration.) (In x)7

A: This is a basic integration problem

Q: Find the indefinite integral. (Use C for the constant of integration.) 5 - ex dx e3x

A:

Q: Find the indefinite integral. (Use C for the constant of integration.) 6 cosh() dx

A: Given that: ∫6cosh(x)xdx

Q: Evaluate the indefinite integral. (Use C for the constant of integration.)

A:

Q: Find the indefinite integral by making a change of variables. ∫x2√(1 − x) dx

A: To determine ∫x21-x dx

Q: Evaluate the indefinite integral. (Use C for the constant of integration.) dx

A:

Q: Evaluate the indefinite integral. (Use C for the constant of integration.) Jove ex V45 + ex dx

A:

Q: Find the indefinite integral. (Use C for the constant of integration.) 5 dx x/ 25 - (In x)2

A: Use substitution method

Q: ou have an opportunity to buy a business that will earn $10,000 per year continuously over the next…

A: To find: The present value of the business. Given information: A business that will earn $10,000 per…

Q: dx 3+Vzx

A: Let  3+√2x = u  √2x = u-3  2x / √2x = u-3  2dx / √2x = du  dx = √2xdu /2  dx = (u-3) du/2  Putting…

Q: What are two options for finding the indefinite integral below? ∫ 5t dt

A: ∫5t dt First option: We can integrate it directly by using the formula ∫xn dx=xn+1n+1+C, we get ∫5t…

Q: Evaluate the indefinite integral. (Use C for the constant of integration.) 2x)9 .

A:

Q: Find the indefinite integral. (Use C for the constant of integration.) 7t + 4 dt

A: ∫e7t+4 dt = e7t+47+Cexplanation is given in below step

Q: Find the Indefinite Integral by making a change of varlab of Integration.) dx

A: Given: I=∫xx-6dx    ....1 Let, u=x-6⇒x=u+6 Now differentiate on both sides with respect to x.…

Q: Find the indefinite integral. (Use C for the constant of integration.) CS dx

A:

Q: Evaluate the indefinite integral. (Use C for the constant of integration.) dx (6x – 5)2

A:

Q: dt

A: We have to find indefinite integral

Q: Evaluate the indefinite integral. (Use C for the constant of integration.) xp (x² + 5)2 х

A:

Q: Find the indefinite integral. (Use C for the constant of integration.) el/x6 xp-

A: integral, ∫e1x6x7dx

Q: Evaluate the indefinite integral. (Use C for the constant of integration.) dx 1 + x20

A:

Q: 3t dt 1 - 9t4

A:

Q: Use integration by parts to evaluate the definite integral te dt.

A:

Q: Find the indefinite integral. (Use C for the constant of integration.) 6 cosh(Vx)

A:

Q: What is the difference between an antiderivate of a function and the indefinite integral of that…

A: An anti-derivative of f(x) is just any function whose derivative is f(x) whereas an indefinite…

Q: 15 50 + n'(t)dt

A:

Q: May you find the definite integral?

A:

Q: The rate of U.S. per capita sales of bottled water for the period 2007–2014 could be approximated by…

A: For 2008, t=2008 - 2007= 1 For 2012, t=2012 - 2007=5   Given that the rate  of U.S. per capita sales…

Q: What does it mean when the integral used to calculate work gives a negative answer?

A:

Q: Find the indefinite integral. (Use C for the constant of integration.) 7 dx 49 - (In x)2

A:

Q: Use the Exponential Rule to find the indefinite integral. (Use C for the constant of integration.)…

A: ∫7xe8x2 dxlet u = 8x2→ du = 16 x dx→  dx = 116xdu= 716∫eu du  Exponential rule :  ∫ax dx = axln (a)…

Q: Find the general indefinite integral. (Use C for the constant of integration.)

A:

Q: Why can the constant of integration be omitted from the antiderivativewhen evaluating a definite…

A: We omit the constant of integration when evaluating a definite integral, because we subtract the…

Q: Evaluate the indefinite integral. (Use C for the constant of integration.) dx 2

A: Given an indefinite integral

Q: How is integration by parts used to evaluate a definite integral?

A: Consider the given statement as How is integration by parts used to evaluate a definite integral

Q: Find the indefinite integral. (Use C for the constant of integration.) x - 2) dx

A: Given,          ∫x8ex9-2dx

Q: Evaluate the indefinite integral. (Use C for the constant of integration.) ex V 33 + eX dx

A:

Q: Find the indefinite integral. (Use C for the constant of integration.) xp

A: The given integral is ∫8x4x2+54dx

Q: Find the indefinite integral. (Use C for the constant of integration.) 7 - ex dx

A:

Q: xp

A:

Q: The left sum, Lpn, is used to approximate the definite integral to the indicated accuracy. How large…

A:

Q: A water quality control agency estimates that a certain factory discharges pollutants into a river…

A:

Q: Find the indefinite integral by making a change of variables. ∫x√(x + 6) dx

A: The given integral is ∫xx+6dx

Q: The rate at which a substance grows is given by R'(x) = 180 e 0.3X, where x is the time (in days).…

A:

Q: Find the indefinite integral. (Use C for the constant of integration.) 3t V9t4

A: Here, we use the formula shown on  board.

Q: when integrating an indefinite integral, your antiderivative must include a "+C" to represent the…

A:

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus. 

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.