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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Compute the trapezoidal approximation T5 to the arc length s of y = tan x over [0, π/4].
Use the Theorem of Pappus to find the volume of the solid of revolution obtained by rotating the region in the first quadrant bounded by y = x2 and y = √x about the y-axis.
Use the Theorem of Pappus to find a formula for the volume of the solid obtained by rotating the triangle with vertices (1, 0), (3, 0), and (2, 2) about the y-axis.
Use the Theorem of Pappus to find the centroid of the half-disk bounded by y = √R2 − x2 and the x-axis.
Sketch the region between y = x + 4 and y = 2 − x for 0 ≤ x ≤ 2. Using symmetry, explain why the centroid of the region lies on the line y = 3. Verify this by computing the moments and the
Calculate the moments and COM of the lamina occupying the region under y = x(4 − x) for 0 ≤ x ≤ 4, assuming a density of ρ = 1200 kg/m3.
Find the centroid of the region lying between the graphs of the functions over the given interval.y = x, y = √x, [0, 1]
Sketch the region between y = 4(x + 1)−1 and y = 1 for 0 ≤ x ≤ 3, and find its centroid.
Find the centroid of the region between the semicircle y = √1 − x2 and the top half of the ellipse y = 1/2 √1 − x2 (Figure 2). -1 y = √₁ - x² y = ²√₁-x² 1 X
Find the centroid of the region lying between the graphs of the functions over the given interval.y = x2, y = √x, [0, 1]
Find the centroid of the shaded region in Figure 5 bounded on the left by x = 2y2 − 2 and on the right by a semicircle of radius 1. Use symmetry and additivity of moments. x = 2y² -
Find the centroid of the region lying between the graphs of the functions over the given interval.y = x−1, y = 2 − x, [1, 2]
Find the centroid of the region lying between the graphs of the functions over the given interval.y = ex, y = 1, [0, 1]
Find the centroid of the region lying between the graphs of the functions over the given interval.y = ln x, y = x − 1, [1, 3]
Find the centroid of the region lying between the graphs of the functions over the given interval.y = sin x, y = cos x, [0, π/4]
Find the centroid of the region. 2 Top half of the ellipse (+)² 1
Sketch the region enclosed by y = x + 1 and y = (x − 1)2 and find its centroid.
Sketch the region enclosed by y = 0, y = (x + 1)3, and y = (1 − x)3, and find its centroid.
Find the centroid of the region.Top half of the ellipse for arbitrary a, b > 0 (²) ²³ + (²)³² ² a b = 1
Find the centroid of the region.Quarter of the unit circle lying in the first quadrant
Find the centroid of the shaded region of the semicircle of radius r in Figure 24. What is the centroid when r = 1 and h = 1/2? Use geometry rather than integration to show that the area of the
Find the centroid of the region.Region between y = x(a − x) and the x-axis for a a > 0
Sketch the region between y = xn and y = xm for 0 ≤ x ≤ 1, where m > n ≥ 0, and find the COM of the region. Find a pair (n,m) such that the COM lies outside the region.
Use the additivity of moments to find the COM of the region.Isosceles triangle of height 2 on top of a rectangle of base 4 and height 3 (Figure 25) -2 2 3 نا 2 * X
Find the formula for the volume of a right circular cone of height H and radius R using the Theorem of Pappus as applied to the triangle bounded by the x-axis, the y-axis, and the line y = −H/R x +
Use the additivity of moments to find the COM of the region.An ice cream cone consisting of a semicircle on top of an equilateral triangle of side 6 (Figure 26) -3 6 +X 3
Let S be the lamina of mass density ρ = 1 obtained by removing a circle of radius r from the circle of radius 2r shown in Figure 27. Let MSx and MSy denote the moments of S. Similarly, let Mbig y
Use the additivity of moments to find the COM of the region. Three-quarters of the unit circle (remove the part in the fourth quadrant)
Find the COM of the laminas in Figure 28 obtained by removing squares of side 2 from a square of side 8. 8 2 8 N
A median of a triangle is a segment joining a vertex to the midpoint of the opposite side. Show that the centroid of a triangle lies on each of its medians, at a distance two-thirds down from the
Prove directly that Eqs. (2) and (3) are equivalent in the following situation. Let ƒ be a positive decreasing function on [0, b] such that ƒ(b) = 0. Set d = ƒ(0) and g(y) = ƒ −1(y). Show
Let P be the COM of a system of two weights with masses m1 and m2 separated by a distance d. Prove Archimedes’s Law of the (weightless) Lever: P is the point on a line between the two weights
Find the COM of a system of two weights of masses m1 and m2 connected by a lever of length d whose mass density ρ is uniform. The moment of the system is the sum of the moments of the weights and
Let R be the region under the graph of y = ƒ(x) over the interval [−a, a], where ƒ(x) ≥ 0. Assume that R is symmetric with respect to the y-axis.(a) Explain why y = ƒ(x) is even—that is, why
Let R be a lamina of uniform density submerged in a fluid of density w (Figure 29). Prove the following law: The fluid force on one side of R is equal to the area of R times the fluid pressure on the
Determine the order of the following differential equations: (a) x³y' = 1 (c) y"" + x¹y' = 2 (b) (y')³ + x = 1 (d) sin(y") + x = y
Which of the following differential equations are directly integrable? (a) y' = x + y dP (c) (e) - dt dx dt = 4P + 1 = 1²e-3t dy dx (b) x- (d) dw 2t dt 1+ 4t = = 3 (f) 12 dx dt = x - 1
Which of the following differential equations are first order? (a) y' = x² (c) (y)³ + yy' = sin x (e) y" + 3y' = y X (b) y = y² (d) x²ye'y (f) yy' + x + y = 0 sin y
Which of the following differential equations are separable? dy (a) = x - 2y dx (c) y' = x²y² dy dt (e) t- = 3√√√1+y (b) xy' + 8ye* = 0 (d) y' = 1-y² dP P+t (f) = dt t
Verify that p(t) = 1/50 e−t/50 satisfies the condition SP 0 p(t)dt = 1
Figure 10 shows the wall of a dam on a water reservoir. Use the Trapezoidal Rule and the width and depth measurements in the figure to estimate the fluid force on the wall. Depth
Let T be the triangular lamina in Figure 23 and assume P = 6.(a) Show that the horizontal cut at height y has length 4 − 2/3 y and use Eq. (2) to compute Mx (with ρ = 1).(b) Use the Symmetry
Approximate the arc length of the curve over the interval using the Trapezoidal Rule TN, the Midpoint Rule MN, or Simpson’s Rule SN as indicated. y = x¹, [1,2], T5
Assume in Figure 10 that the depth of water in the reservoir dropped 20 ft in a drought. Use the Trapezoidal Rule and the measurements in the figure to estimate the fluid force on the wall. Depth
Verify that for all r > 0, the exponential density function p(t) = 1/r e−t/r satisfies the condition 00€ SP 0 p(t)dt = 1
Approximate the arc length of the curve over the interval using the Trapezoidal Rule TN, the Midpoint Rule MN, or Simpson’s Rule SN as indicated. y = sinx, [0,1, M8
Let T be the triangular lamina in Figure 23, and assume P = 8 and ρ = 1. Find the center of mass.
Calculate the surface area of the solid obtained by rotating the curve over the given interval about the x-axis.y = x + 1, [0, 4]
Calculate the fluid force on a side of the plate in Figure 11(A), submerged in water, assuming that the top of the plate is at a depth of D =2 m. D 2 m 4 m 7m (A) 2 m 4 m 2 m (B)
Approximate the arc length of the curve over the interval using the Trapezoidal Rule TN, the Midpoint Rule MN, or Simpson’s Rule SN as indicated. -1 y = x¹, [1,2], S8
Calculate the surface area of the solid obtained by rotating the curve over the given interval about the x-axis. 2 y = 23²,43/4 - 3²-15/4² [0, 1]
Approximate the arc length of the curve over the interval using the Trapezoidal Rule TN, the Midpoint Rule MN, or Simpson’s Rule SN as indicated. y = ex², [0,2], S8 e-x²₁
Calculate the fluid force on a side of the plate in Figure 11(A), submerged in water, assuming that the top of the plate is at a depth of D =4 m.
Find the centroid of the region lying underneath the graph of the function over the give interval.ƒ(x) = √x, [1, 4]
Calculate the surface area of the solid obtained by rotating the curve over the given interval about the x-axis. y = 2 3 3/2 1 2 x¹/², [1,2]
Approximate the arc length of the curve over the interval using the Trapezoidal Rule TN, the Midpoint Rule MN, or Simpson’s Rule SN as indicated.y = ln x, [1, 3], M6
Calculate the fluid force on a side of the plate in Figure 11(B), submerged in a fluid of mass density ρ = 800 kg/m3. D 2 m 4 m 7m (A) 2m 4 m 2 m (B)
Find the centroid of the region lying underneath the graph of the function over the give interval.ƒ(x) = x3, [0, 1]
Calculate the surface area of the solid obtained by rotating the curve over the given interval about the x-axis. y = 1 2x², [0,2]
Approximate the arc length of the curve over the interval using the Trapezoidal Rule TN, the Midpoint Rule MN, or Simpson’s Rule SN as indicated.y = cos x, [0, 2], T8
Find the fluid force on the side of the plate in Figure 12, submerged in a fluid of density ρ = 1200 kg/m3. The top of the plate is level with the fluid surface. The edges of the plate are the
Compute the total surface area of the coin obtained by rotating the region in Figure 1 about the x-axis. The top and bottom parts of the region are semicircles with a radius of 1 mm. 1 mm 4 mm -x
Find the centroid of the region lying underneath the graph of the function over the give interval.ƒ(x) = 9 − x2, [0, 3]
Let R be the plate in the shape of the region under y = sin x for 0 ≤ x ≤ π/2 in Figure 13(A). If R is rotated counterclockwise by 90◦ and then submerged in a fluid of density 1100 kg/m3 with
Approximate the arc length of the curve over the interval using the Trapezoidal Rule TN, the Midpoint Rule MN, or Simpson’s Rule SN as indicated.y = x sin x, [0, 10π], T100
F(z) denotes the cumulative normal distribution function. Refer to a calculator or computer algebra system to obtain values of F(z).Express the area of region A in Figure 9 in terms of F(z) and
Find the centroid of the region lying underneath the graph of the function over the give interval.ƒ(x) = (1 + x2)−1/2, [0, 3]
Approximate the arc length of the curve over the interval using the Trapezoidal Rule TN, the Midpoint Rule MN, or Simpson’s Rule SN as indicated. y = [0, 0.99], S100
Calculate the fluid force on the side of a right triangle of height 3 m and base 2 m submerged in water vertically, with its upper vertex at the surface of the water.
Find the centroid of the region lying underneath the graph of the function over the give interval.ƒ (x) = e−x, [0, 4]
Calculate the fluid force on the side of a right triangle of height 3 m and base 2 m submerged in water vertically, with its upper vertex located at a depth of 4 m.
A plate in the shape of the shaded region in Figure 2 is submerged in water. Calculate the fluid force on a side of the plate if the water surface is y = 1. - 1 y=√1 y = √₁-x² + ²y = ²/²
Find the centroid of the region lying underneath the graph of the function over the give interval.ƒ(x) = ln x, [1, 2]
Figure 3 shows an object whose face is an equilateral triangle with 5-m sides. The object is 2 m thick and is submerged in water with its vertex 3 m below the water surface. Calculate the fluid force
The end of a horizontal oil tank is an ellipse (Figure 4) with equation (x/4)2 + (y/3)2 = 1 (length in meters). Assume that the tank is filled with oil of density 900 kg/m3.(a) Calculate the total
Find the centroid of the region lying underneath the graph of the function over the give interval.ƒ(x) = sin x, [0, π]
Calculate the moments and center of mass of the lamina occupying the region between the curves y = x and y = x2 for 0 ≤ x ≤ 1.
In the notation of Exercise 17, calculate the fluid force on a side of the plate R if it is oriented as in Figure 13(A).Data From Exercise 17Let R be the plate in the shape of the region under y =
F(z) denotes the cumulative normal distribution function. Refer to a calculator or computer algebra system to obtain values of F(z).Show that the area of region B in Figure 9 is equal to 1 − F(1.5)
Calculate the length of the astroid x2/3 + y2/3 = 1 (Figure 14). -1 1 +1-
Calculate the fluid force on one side of a plate in the shape of region A shown in Figure 14. The water surface is at y = 1, and the fluid has density ρ = 900 kg/m3. - y B1 y = ln x A e X
Calculate the fluid force on one side of the “infinite” plate B in Figure 14, assuming the fluid has density ρ = 900 kg/m3. 1 y B 1 y = ln x A e X
F(z) denotes the cumulative normal distribution function. Refer to a calculator or computer algebra system to obtain values of F(z).Assume X has a standard normal distribution (μ = 0, σ = 1).
Show that the arc length of the astroid x2/3 + y2/3 = a2/3 (for a > 0) is proportional to a.
Figure 15(A) shows a ramp inclined at 30◦ leading into a swimming pool. Calculate the fluid force on the ramp. Water surface 6 (A) 30⁰ 4
F(z) denotes the cumulative normal distribution function. Refer to a calculator or computer algebra system to obtain values of F(z).Assume X has a normal distribution with μ = 0 and σ = 5. Express
F(z) denotes the cumulative normal distribution function. Refer to a calculator or computer algebra system to obtain values of F(z).Use a graph to show that F(−z) = 1 − F(z) for all z. Then show
Find the length of the arc of the curve x2 = (y − 2)3 from P(1, 3) to Q(8, 6).
Find the arc length of the curve shown in Figure 15. 0.5 y 1 2 +x 3
The massive Three Gorges Dam on China’s Yangtze River has height 185 m (Figure 16). Calculate the force on the dam, assuming that the dam is a trapezoid of base 2000 m and upper edge 3000 m,
F(z) denotes the cumulative normal distribution function. Refer to a calculator or computer algebra system to obtain values of F(z).The average September rainfall in Erie, Pennsylvania, is a random
Find the value of a such that the arc length of the catenary y = cosh x for −a ≤ x ≤ a equals 10.
Calculate the arc length of the graph of ƒ(x) = mx + r over [a, b] in two ways: using the Pythagorean Theorem (Figure 16) and using the arc length integral. y + a b-a m(b-a) b
A bottling company produces bottles of fruit juice that are filled, on average, with 32 ounces of juice. Due to random fluctuations in the machinery, the actual volume of juice is normally
A square plate of side 3 m is submerged in water at an incline of 30° with the horizontal. Calculate the fluid force on one side of the plate if the top edge of the plate lies at a depth of 6 m.
According to Maxwell’s Distribution Law, in a gas of molecular mass m, the speed v of a molecule in a gas at temperature T (kelvins) is a random variable with densitywhere k is Boltzmann’s
The trough in Figure 18 is filled with corn syrup, whose weight density is 90 lb/ft3. Calculate the force on the front side of the trough. h b Buch. a
Show that the circumference of the unit circle is equal toEvaluate, thus verifying that the circumference is 2π. 25₁ dx √1-x² (an improper integral)
Define the median of a probability distribution to be that value a such thatShow that if a probability function is symmetric about the line x = m, then m is both the mean and the median. SPC = [_p(x)
Calculate the fluid pressure on one of the slanted sides of the trough in Figure 18 when it is filled with corn syrup as in Exercise 25.Data From Exercise 25The trough in Figure 18 is filled with

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