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calculus

Questions and Answers of Calculus

The velocity of an object is v(t) = 2 - |t - 2|. Assuming that the object is at the origin at time 0, find a formula for its position at time t. When, if ever, does the object return to the origin?
The velocity of an object is(a) Assuming that the object is at the origin at time 0, find a formula for its position at time t(t‰¥0) (b) What is the farthest to the right of the origin that
Let f be continuous on [a, b] and thus integrable there. Show that
Suppose that f' is integrable and |f'(x)| ≤ M for all x. Prove that |f(x)| ≤ |f(a)| +M|x -a| for every a.
Suppose thatAnd Use properties of definite integrals interval additively, and so on to calculate each of the integrals in problems. (a) (b) (c)
In problems, use the Second Fundamental Theorem of Calculus to evaluate each definite integral(a)(b) (c)
In problems, use the method of substitution to find each of the following indefinite integrals.(a)(b) (c) (d)
In Problems, use the Substitution Rule for Definite Integrals to evaluate each definite integral.(a)(b) (c)
Figure 4 shows the graph of a function f that has a continuous third derivative. The dashed lines are tangent to the graph of y = f(x) at the points (0, 2) and (3, 0), Based on what is shown, tell,
Figure 5 shows the graph of a function f that has a continuous third derivative. The dashed lines are tangent to the graph of y = f(x) at the points (0, 2) and (4, 1). Based on what is shown, tell,
Water leaks out of a 200-gallon storage tank (initially full) at the rate V'(t) = 20 - t, where t is measured in hours and V in gallons. How much water leaked out between 10 and 20 hours? How long
Oil is leaking at the rate of V' (t) = 1 - t/110 from a storage tank that is initially full of 55 gallons. How much leaks out during the first hour? During the tenth hour how long until the entire
The water usage in a small town is measured in gallons per hour. A plot of this rate of usage is shown in Figure 6 for the hours midnight through noon for a particular day. Estimate the total amount
Figure 7 shows the rate of oil consumption in million barrels per year for the United States from 1973 to 2003. Approximately how many barrels of oil were consumed between 1990 and 2000?
Figure 8 shows the power usage, measured in megawatts, for a small town for one day (measured from midnight to mid-night). Estimate the energy usage for the day measured in megawatt-hours.
We claim that(a) Use Figure 9 to justify this by a geometric argument. (b) Prove the result using the Second Fundamental Theorem of Calculus. (c) Show that An = nBn
Prove the Second Fundamental Theorem of Calculus following the method suggested in Example 6 of section 4.3
In Problems, first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus(a)(b) (c)
Explain why (1/n3)Should be a good approximation to For large n. Now calculate the summation expression for n = 10, and evaluate the integral by the Second Fundamental Theorem of Calculus. Compare
Show that 1/2x |x| is an anti-derivative of |x|, and use this fact to get a simple formula for
Find a nice formula for
Give an example to show that the accumulation functionCan be continuous even if f is not continuous
In problems, find the average value of the function on the given interval. (a). f(x) = 4x3; [1, 3] (b). f(x) = 5x2; [1, 4] (c). f(x) = x/√x2 + 16; [0, 3] (d). f(x) = x/√x3 + 16; [0, 2]
In problems, find all values of c that satisfy the Mean Value Theorem for integrals on the given interval. (a) f(x) = √(x + 1); [0,3] (b) f(x) = x2; [-1,1] (c) f(x) = 1 - x2; [-4,3] (d) f(x) = x(1
Use a graphing calculator to plot the graph of the integrand in Problems. Then estimate the integral as suggested in the margin note accompanying Theorem B.(a)(b) (c) (d)
Figure 11 shows temperature T as a function of time t (measured in hours past midnight) for one day in St. Louis, Missouri.(a) Approximate the average temperature for the day.(b) Must there be a time
In Problems, use symmetry to help you evaluate the given integral.(a)(b) (c) (d)
How doesCompare with When f is an even function? An odd function
Prove (by a substitution) that
If f is periodic with period p, thenConvince yourself that this is true by drawing a picture and then use the result to calculate
Prove or disprove that the integral of the average value equals the integral of the function on the intervalWhere f is the average value of the function f is the average value of the function f over
Assuming that u and v can be integrated over the interval [a, b] and that the average values over the interval are denoted by u and v, prove or disprove that(a).(b) Where k is any constant; (c). if u
Give a proof of the mean value Theorem for Integrals (Theorem A) that does not use the First Fundamental Theorem of calculus.
Integrals that occur frequently in applications are(a) Using a trigonometric identity, show that (b) Show from graphical consideration that (c) Conclude that
Complete the generalization of the Pythagorean theorem begun in Problem 59 of section 0.3 by showing that A+B = C in figure 12 these being the areas of similar regions built on the two legs and the
Prove the symmetry Theorem for the case of odd functions.
In Problems, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with n = 8 to approximate the definite integral Then use the Second Fundamental
In problems, determine an n so that the Trapezoidal Rule will approximate the integral with an error En satisfying |En| ‰¤ 0.01. Then, using that n, approximate the integral(a).(b).
In Problems, determine an n so that the Parabolic Rule will approximate the integral with an error En satisfying |En| ‰¤ 0.01. Then, using that n, approximate the integral(a)(b)
Let f(x) = ax2 + bx + c. show thatBoth have the value (h/3)[a(6m2+2h2) + b(6m) + 6c]. This establishes the area formula on which the Parabolic Rule is based.
Show that the Parabolic Rule is exact for any cubic polynomial in two different ways. (a) By direct calculation. (b) By showing that En = 0.
Justify your answers to Problems two ways: (1) using the properties of the graph of the function, and (2) using the error formulas from Theorem A.(a) If a function f is increasing on [a, b], will the
Show that the parabolic Rule gives the exact value ofProvided that k is odd.
It is interesting that a modified version of the Trapezoidal Rule turns out to be in general more accurate than the Parabolic Rule. This version says thatWhere T is the standard trapezoidal
Figure 8 shows the depth in feet of the water in a river measured at 20-foot intervals across the width of the river. If the river flows at 4 miles per hour, how much water (in cubic feet) flows past
On her way to work, Teri noted her speed every 3 minutes. The results are shown in the table below. How far did she drive?
In Problems, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) midpoint Riemann sum, (4) Trapezoidal Rule, (5) Parabolic Rule with n = 4, 8, 16. Present your approximations in a
In problems, use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral (or integrals) for the area of the indicated region.(a)(b) (b)
In Problems, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of
Sketch the region R bounded by y = x + 6, y = x3 and 2y + x = 0. Then find its area.
Find the area of the triangle with vertices at (-1. 4), (2, -2), and (5, 1) by integration.
An object moves along a line so that its velocity at time t is v(t) = 3t2 - 24t + 36 feet per second. Find the displacement and total distance traveled by the object for -1 ≤ t ≤ 9.
Follow the directions of Problem 31 if v (t) = 1/2 + sin 2t and the interval is 0 ≤ t ≤ 3π/2.
Starting at s = 0 when t = 0, an object moves along a line so that its velocity at time t is v(t) = 2t - 4 centimeters per second. How long will it take to get to s = 12? To travel a total distance
Consider the curve y= 1/x2 for 1 ≤ x ≤ 6. (a) Calculate the area under this curve. (b) Determine c so that the line x = c bisects the area of part (a). (c) Determine d so that the line y = d
Calculate areas A, B, C and D in Figure 12. Check by calculating A+B+C+D in one integration
Prove Cavalieri's Principle. (Bonaventura Cavalieri (1598-1647) developed this principle in 1635.) If two regions have the same height at every x in [a, 1th then they have the same area (see Figure
Use Cavalieri's Principle (not integration see problem 36) to show that the shaded region in figure have same area.
In problems, find the volume of the solid generated when the indicated region is revolved about the specified axis; slice, approximate, integrate.(a) x-axis(b) x-axis
In Problems, sketch the region R bounded by the graphs of the given equations and show a typical horizontal slice. Find the volume of the solid generated by revolving R about the y-axis (a) x = y2, x
Find the volume of the solid generated by revolving about the x-axis the region bounded by the upper half of the ellipse x2/a2 + y2/b2 = 1 and the x-axis, and thus find the volume of a prolate
Find the volume of the solid generated by revolving about the x-axis the region bounded by the line y = 6x and the parabola y = 6x2.
Find the volume of the solid generated by revolving about the x-axis the region bounded by the line x - 2y = 0 and the parabola y2 = 4x.
Find the volume of the solid generated by revolving about the x-axis the region in the first quadrant bounded by the circle x2 + y2 = r2, the x-axis, and the line x = r - h, 0 < h < r, and thus find
Find the volume of the solid generated by revolving about the y-axis the region bounded by the line y = 4x and the parabola y = 4x2.
Find the volume of the solid generated by revolving about the line y = 2 the region in the first quadrant bounded by the parabolas 3x2 - 16y + 48 = 0 and x2 - 16y + 80 = 0 and the y-axis.
The base of a solid is the region inside the circle x2 + y2 = 4. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a square.
The base of a solid is bounded by one arch of y = √cos x, -π/2 ≤ x ≤ π/2, and the x-axis. Each cross section perpendicular to the x-axis is a square sitting on this base. Find the volume of
The base of a solid is the region bounded by y = 1 - x2 and y = 1 - x4. Cross sections of the solid that are perpendicular to the x-axis are squares. Find the volume of the solid.
Find the volume of one octant (one-eighth) of the solid region common to two right circular cylinders of radius 1 whose axes intersect at right angles.
Find the volume inside the "+" shown in Figure 16, Assume that both cylinders have radius 2 inches and length 12 inches
Find the volume inside the "+" shown in Figure 16 assuming the both cylinders have radius r and length L
Find the volume inside the "T" in Figure 17, assuming that each cylinder has radius r = 2 inches and that the lengths are L1 = 12 inches and L2 = 8 inches.
Repeat Problem 30 for arbitrary r, L1, and L2.In problem 30Find the volume inside the "T" in Figure 17, assuming that each cylinder has radius r = 2 inches and that the lengths are L1 = 12 inches and
The base of a solid is the region R bounded by y = √x and y = x2. Each cross section perpendicular to the x-axis is a semicircle with diameter extending across R. Find the volume of the solid.
Find the volume of the solid generated by revolving the region in the first quadrant bounded by the curve y2 = x3, the line x = 4, and the x-axis: (a) About the line x = 4; (b) About the line y = 8.
Find the volume of the solid generated by revolving the region bounded by the curve y2 = x3, the line y = 8, and the y-axis: (a) About the line x = 4; (b) About the line y = 8.
Complete the evaluation of the integral in Example 4 by noting thatNow interpret the first integral as the area of a quarter circle.
An open barrel of radius r and height It is initially full of water. It is tilted and water pours out until the water level coincides with a diameter of the base and just touches the rim of the top.
A wedge is cut from a right circular cylinder of radius r (Figure 19). The upper surface of the wedge is in a plane through a diameter of the circular base and makes an angle 0 with the base. Find
A water tank is obtained by revolving the curve y = kx4, k > 0, about the y-axis. (a) Find V (y), the volume of water in the tank as a function of its depth y. (b) Water drains through a small hole
Show that the volume of a general cone (Figure 20) is 5 Ah, where A is the area of the base and h is the height. Use this result to give the formula for the volume of(a) A right circular cone of
In Problems, sketch the region R bounded by the graphs of the given equations and show a typical vertical slice. Then find the volume of the solid generated by revolving R about the x-axis (a) y =
In Problem, find the volume of the solid generated when the region R bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps (a) Sketch the region
Consider the region R (Figure 8). Set up an integral for the volume of the solid obtained when R is revolved about the given line using the indicated method.(a) The x-axis (washers)(b) The y-axis
A region R is shown in Figure 9. Set up an integral for the volume of the solid obtained when R is revolved about each of the following lines. Use the indicated method.(a) The y-axis (washers)(b) The
Sketch the region R bounded by y = 1/x3, x = 1, x = 3, and y = 0. Set up (but do not evaluate) integrals for each of the following (a) Area of R (b) Volume of the solid obtained when R is revolved
Follow the directions of Problem 15 for the region R bounded by y = x3 + 1 and y = 0 and between x = 0 and x = 2. In problem 15 (a) Area of R (b) Volume of the solid obtained when R is revolved
Find the volume of the solid generated by revolving the region R bounded by the curves x = √y and x = y3/32 about the x-axis.
Follow the directions of Problem 17, but revolve R about the line y = 4. In problem 17 Find the volume of the solid generated by revolving the region R bounded by the curves x = √y and x = y3/32
A round hole of radius a is drilled through the center of a solid sphere of radius b (assume that b > a). Find the volume of the solid that remains.
Set up the integral (using shells) for the volume of the torus obtained by revolving the region inside the circle x2 + y2 = a2 about the line x = b, where b > a. Then evaluate this integral.
The region in the first quadrant bounded by x = 0, y = sin (x2), and y = cos(x2) is revolved about the y-axis. Find the volume of the resulting solid.
The region bounded by y = 2 + sin x, y = 0, x = 0, and x = 2a is revolved about the y-axis. Find the volume those results.
Let R be the region bounded by y = x2 and y = x. Find the volume of the solid that results when R is revolved around: (a) The x-axis; (b) The y-axis; (c) The line y = x.
In problems, find the length of the indicated curve. a. y = 4x3/2 between x = 1/3 and x = 5 b. y = 2/3(x2 + 1)3/2 between x = 1 and x = 2
Use an x-integration to find the length of the segment of the line y = 2x + 3 between x = 1 and x = 3. Check by using the distance formula.
Use a y-integration to find the length of the segment of the line 2y - 2x + 3 = 0 between y = 1 and y = 3. Check by using the distance formula.
In Problems, set up a definite integral that gives the arc length of the given curve Approximate the integral using the Pam-bulk Rule with n = 8. (a) x = t, y = t2; 0 ≤ t ≤ 2 (b) x = t2, y =
Sketch the graph of the four-cusped hypocycloid x = a sin3 t, y = a cos3 t, 0 ≤ t ≤ 2π, and find its length. Hint: By symmetry, you can quadruple the length of the first quadrant portion.

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