A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.9 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. Find the work done. (Use 9.8 m/s? for g.) Show how to approximate the required work by a Riemann sum. (Let x be the height in meters above the ground. Enter x,* as x-) ) ax lim Express the work as an integral. 12 ) dx Evaluate the integral. (Round your answer to the nearest integer.)

Q: A bucket that weighs 4 Ib and a rope of negligible weight are used to draw water from a well that is…

A: Let x be the height in feet above the bottom of the well.

Q: A car is entering the highway, and accelerates in the first 30 seconds, as follows: time (in…

A: Given that Time(in seconds) 0 6 12 18 24 30 Speed(in m/s) 2 10 16 20 23 25 To find the total…

Q: Using the left-endpoint Riemann sum with 4 equal subintervals, ed above by the curve f(x) = 1 and…

A:

Q: A bucket that weighs 4 Ib and a rope of negligible weight are used to draw water from a well that is…

A:

Q: Using the link between the Riemann sum and definite integral we se 70 n dx , lim n- co (n+1)2

A: Given: To explain the given statement as follows, Here, limn→∞ ∑i=1n 70 n(n + i)2

Q: | (2² – 4x). (2* — 4г) dx 1

A:

Q: A heavy rope, 60 ft long, weighs 0.7 lb/ft, and hangs over the edge of a building 150 ft high. (Let…

A: We will find the the required work by a Riemann sum.

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A:

Q: Approximate the area between the graph of f(x)=x^3 and the x-axis on the interval [3,5] by using a…

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Q: Use a Riemann sum to approximate (x + 6) dx by using five subintervals of -2 equal width, and taking…

A:

Q: Calculate the left Riemann sum for the given function over the given interval, using the given value…

A:

Q: Estimate / (-In (-a³ +x² + 4)) dx using a left-hand Riemann sum and 6 subintervals of equal width.…

A:

Q: Find the Riemann sum for  f(x) = 2x − 1,    −6 ≤ x ≤ 4,  with five equal subintervals, taking the…

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Q: Calculate the left Riemann sum for the given function over the given interval, using the given value…

A:

Q: Find (x³ + 2x²) dx using the Riemann sum.

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Q: A bucket that weighs 6 Ib and a rope of negligible weight are used to draw water from a well that is…

A:

Q: Use hiemann sum with 3 terms and the midpolnt rule to approximate Hhe value at 13 Sin (x)dx

A:

Q: Calculate the left Riemann sum for the given function over the given interval using the given value…

A: We have to find the left Riemann sun of given function.

Q: Calculate the left Riemann sum for the given function over the given interval using the given value…

A:

Q: Find the formula for the Riemann sum obtained by dividing the interval [0, 1] inton equal…

A: Introduction: Riemann integration is very significant in real analysis. It is an advanced study on…

Q: Calculate the left Riemann sum for the given function over the given interval using the given value…

A: Given function is fx=9x2 Here, n=4 and interval is −2,2 Δx=b−an=2−−24=1 Hence, Δx=1

Q: Consider a function y = k(x) such that k(x) takes on both positive and negative values. What is the…

A: y = k(x) Since k(x) can have both positive and negative value hence, the function goes above and…

Q: Calculate the left Riemann sum for the given function over the given interval using the given value…

A:

Q: A cable that weighs 8 lb/ft is used to lift 750 lb of coal up a mine shaft 600 ft deep. Find the…

A:

Q: A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is…

A:

Q: f(x) = 124x − 31 over [0, 2], n = 4 Calculate the left Riemann sum for the given function over the…

A:

Q: AB5: Use a left Riemann sum with the three subintervals indicated by the table above to approximate…

A: Riemann sum is a method to approximate a definite integral. There are two different kinds of…

Q: A heavy rope, 60 ft long, weighs 0.7 lb/ft and hangs over the edge of a building 130 ft high. (Let x…

A:

Q: Calculate the left Riemann sum for the given function over the given interval using the given value…

A: Obtain the left Riemann sum for the function f(x)=5e-x over 0,10 and n=5 as follows. The left…

Q: A cable that weighs 3 Ib/ft is used to lift 500 lb of coal up a mine shaft 400 ft deep. Find the…

A:

Q: Consider the integral 5² (4x² + 3x + 4) dx 2 a) Find the Riemann sum using right end points and n=3…

A:

Q: If we are using a left Riemann sum (with rectangles of equal width), then what formula does xi…

A: Left riemann sum of any function, defined on a closed interval, represents the sum of area of each…

Q: A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is…

A: To find the work done in pulling the bucket to the top of the well.

Q: Find the formula for the Riemann sum obtained by dividing the interval [ - 1, 0] into n equal…

A:

Q: (b) Use a left Riemann sum with the three subintervals indicated by the data in the table to…

A: Left reimann sum

Q: 1) Approximate (3+x-x²) dx with a left-endpoint Riemann sum using four equal subintervals. 2)…

A:

Q: b) Approximate [₁₂²² + y²dA where D is region inside the upper unit semi-circle centered at the…

A:

Q: (a) Find an approximation to the integral  ∫04      (x2 -3x) dxusing a Riemann sum with right…

A:

Q: A heavy rope, 40 ft long, weighs 0.3 lb/ft and hangs over the edge of a building 90 ft high. (Let x…

A: Our aim is to find the work  done(W) in pulling half the rope to the top of the building.

Q: The integral (16x² + 96x + 144)dx is approximated by the following left endpoint Riemann sum (each…

A:

Q: Calculate the left Riemann sum for the given function over the given interval using the given value…

A:

Q: he graph of the velocity of a car accelerating from rest to a speed of 60 km/h over a period of 30…

A: The graph is:

Q: 1. Approximate |(x² +e*) dx using a Riemann Sum with n=6 and right endpoints. Round result to the…

A: We have ∫06x2+exdx with n=6

Q: | 2.5 2 に に」N に」す の

A: Since, We can see that the no. of subdivisions(or rectangles) which can be made in the patio,…

Q: Calculate the left Riemann sum for the given function over the given interval using the given value…

A: Let f: D → R be a function defined on a subset D of the real line R. Let I = [a, b] be a closed…

Q: Calculate the left Riemann sum for the given function over the given interval using the given value…

A: Use the formula for the left Riemann Sum and simplify to get the answer.

Q: If f(x)= sqrt(x) - 2 and 1 is less than or equal to x which is less than or equal to 6, find the…

A: Formula: The Riemann sum of a function f(x) over a interval a, busing n rectangles is given by,…

Q: The graph of the velocity of a braking car is shown. v (ft/s) 75 50 25 t (seconds) 2 4 Use this…

A: To estimate the distance traveled by the car using the given graph and midpoint Riemann sum.

Q: A bucket that weighs 4 Ib and a rope of negligible weight are used to draw water from a well that is…

A: Given

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.