Stewart 8th Section 8.3: Applications to Physics and Engineering

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Stewart 8th Section 8.3: Applications to Physics and Engineering 0/58
1 Hydrostatic Pressure and Force · Level 1
An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find (a) the hydrostatic pressure on the bottom of the aquarium, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the aquarium.
2 Hydrostatic Pressure and Force · Level 2
A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density \(820\) kg/m\(^3\) to a depth of \(1.5\) m. Find (a) the hydrostatic pressure on the bottom of the tank, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the tank.
3 Hydrostatic Force · Level 2
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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4 Hydrostatic Force · Level 2
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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5 Hydrostatic Force · Level 2
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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6 Hydrostatic Force · Level 2
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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7 Hydrostatic Force · Level 3
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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8 Hydrostatic Force · Level 3
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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9 Hydrostatic Force · Level 3
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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10 Hydrostatic Force · Level 3
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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11 Hydrostatic Force · Level 3
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
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12 Hydrostatic Force · Level 3
A milk truck carries milk with density \(64.6\) lb/ft\(^3\) in a horizontal cylindrical tank with diameter \(6\) ft.
(a) Find the force exerted by the milk on one end of the tank when the tank is full.
(b) What if the tank is half full?

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13 Hydrostatic Force · Level 3
A trough is filled with a liquid of density \(840\) kg/m\(^3\). The ends of the trough are equilateral triangles with sides \(8\) m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.
14 Hydrostatic Force · Level 3
A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.
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15 Hydrostatic Force · Level 2
A cube with \(20\)-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Find the hydrostatic force on (a) the top of the cube and (b) one of the sides of the cube.
16 Hydrostatic Force · Level 4
A dam is inclined at an angle of \(30^{\circ}\) from the vertical and has the shape of an isosceles trapezoid \(100\) ft wide at the top and \(50\) ft wide at the bottom and with a slant height of \(70\) ft. Find the hydrostatic force on the dam when it is full of water.
17 Hydrostatic Force · Level 3
A swimming pool is \(20\) ft wide and \(40\) ft long and its bottom is an inclined plane, the shallow end having a depth of \(3\) ft and the deep end, \(9\) ft. If the pool is full of water, find the hydrostatic force on (a) the shallow end, (b) the deep end, (c) one of the sides, and (d) the bottom of the pool.
18 Hydrostatic Force · Level 3
Suppose that a plate is immersed vertically in a fluid with density \(\rho\) and the width of the plate is \(w(x)\) at a depth of \(x\) meters beneath the surface of the fluid. If the top of the plate is at depth \(a\) and the bottom is at depth \(b\), show that the hydrostatic force on one side of the plate is \(F = \displaystyle\int_{a}^{b} \rho g x w(x) d x\).
19 Hydrostatic Force (Simpson's Rule) · Level 3
A metal plate was found submerged vertically in seawater, which has density \(64\) lb/ft\(^3\). Measurements of the width of the plate were taken at the indicated depths. Use Simpson's Rule to estimate the force of the water against the plate. Depth (m): \(7.0\), \(7.4\), \(7.8\), \(8.2\), \(8.6\), \(9.0\), \(9.4\) Plate width (m): \(1.2\), \(1.8\), \(2.9\), \(3.8\), \(3.6\), \(4.2\), \(4.4\)
20 Hydrostatic Force and Centroid · Level 4
(a) Use the formula of Exercise 18 to show that \(F = (\rho g \bar{x})A\) where \(\bar{x}\) is the \(x\)-coordinate of the centroid of the plate and \(A\) is its area. This equation shows that the hydrostatic force against a vertical plane region is the same as if the region were horizontal at the depth of the centroid of the region.
(b) Use the result of part (a) to give another solution to Exercise 10.

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21 Moments and Center of Mass (1D) · Level 1
Point-masses \(m_i\) are located on the \(x\)-axis as shown. Find the moment \(M\) of the system about the origin and the center of mass \(\bar{x}\). \(m_1 = 6\) at \(x = 10\), \(m_2 = 9\) at \(x = 30\).
22 Moments and Center of Mass (1D) · Level 1
Point-masses \(m_i\) are located on the \(x\)-axis as shown. Find the moment \(M\) of the system about the origin and the center of mass \(\bar{x}\).
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23 Moments and Center of Mass (2D) · Level 2
The masses \(m_i\) are located at the points \(P_i\). Find the moments \(M_x\) and \(M_y\) and the center of mass of the system. \(m_1 = 4\), \(m_2 = 2\), \(m_3 = 4\); \(P_1(2, -3)\), \(P_2(-3, 1)\), \(P_3(3, 5)\).
24 Moments and Center of Mass (2D) · Level 2
The masses \(m_i\) are located at the points \(P_i\). Find the moments \(M_x\) and \(M_y\) and the center of mass of the system. \(m_1 = 5\), \(m_2 = 4\), \(m_3 = 3\), \(m_4 = 6\); \(P_1(-4, 2)\), \(P_2(0, 5)\), \(P_3(3, 2)\), \(P_4(1, -2)\).
25 Centroid of Region · Level 2
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. \(y = 2x\), \(y = 0\), \(x = 1\)
26 Centroid of Region · Level 2
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. \(y = \sqrt{x}\), \(y = 0\), \(x = 4\)
27 Centroid of Region · Level 2
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. \(y = e^x\), \(y = 0\), \(x = 0\), \(x = 1\)
28 Centroid of Region · Level 2
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. \(y = \sin x\), \(y = 0\), \(0 \leq x \leq \pi\)
29 Centroid of Region · Level 3
Find the centroid of the region bounded by the given curves. \(y = x^2\), \(x = y^2\)
30 Centroid of Region · Level 3
Find the centroid of the region bounded by the given curves. \(y = 2 - x^2\), \(y = x\)
31 Centroid of Region · Level 3
Find the centroid of the region bounded by the given curves. \(y = \sin x\), \(y = \cos x\), \(x = 0\), \(x = \dfrac{\pi}{4}\)
32 Centroid of Region · Level 3
Find the centroid of the region bounded by the given curves. \(y = x^3\), \(x + y = 2\), \(y = 0\)
33 Centroid of Region · Level 3
Find the centroid of the region bounded by the given curves. \(x + y = 2\), \(x = y^2\)
34 Moments and Center of Mass (Lamina) · Level 3
Calculate the moments \(M_x\) and \(M_y\) and the center of mass of a lamina with the given density and shape. \(\rho = 4\)
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35 Moments and Center of Mass (Lamina) · Level 3
Calculate the moments \(M_x\) and \(M_y\) and the center of mass of a lamina with the given density and shape. \(\rho = 6\)
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36 Centroid (Simpson's Rule) · Level 3
Use Simpson's Rule to estimate the centroid of the region shown.
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37 Centroid of Region · Level 3
Find the centroid of the region bounded by the curves \(y = x^3 - x\) and \(y = x^2 - 1\). Sketch the region and plot the centroid to see if your answer is reasonable.
38 Centroid (Numerical) · Level 3
Use a graph to find approximate \(x\)-coordinates of the points of intersection of the curves \(y = e^x\) and \(y = 2 - x^2\). Then find (approximately) the centroid of the region bounded by these curves.
39 Centroid (Proof) · Level 4
Prove that the centroid of any triangle is located at the point of intersection of the medians. [Hints: Place the axes so that the vertices are \((a, 0)\), \((0, b)\), and \((c, 0)\). Recall that a median is a line segment from a vertex to the midpoint of the opposite side. Recall also that the medians intersect at a point two-thirds of the way from each vertex (along the median) to the opposite side.]
40 Centroid (Additivity of Moments) · Level 3
Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments.
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41 Centroid (Additivity of Moments) · Level 3
Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments.
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42 Centroid of Region · Level 4
A rectangle \(cal(R)\) with sides \(a\) and \(b\) is divided into two parts \(cal(R)_1\) and \(cal(R)_2\) by an arc of a parabola that has its vertex at one corner of \(cal(R)\) and passes through the opposite corner. Find the centroids of both \(cal(R)_1\) and \(cal(R)_2\).
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43 Centroid (Proof) · Level 3
If \(\bar{x}\) is the \(x\)-coordinate of the centroid of the region that lies under the graph of a continuous function \(f\), where \(a \leq x \leq b\), show that \(\displaystyle\int_{a}^{b} (c x + d) f(x) d x = (c \bar{x} + d) \displaystyle\int_{a}^{b} f(x) d x\).
44 Theorem of Pappus · Level 2
Use the Theorem of Pappus to find the volume of the given solid. A sphere of radius \(r\) (Use Example 4.)
45 Theorem of Pappus · Level 3
Use the Theorem of Pappus to find the volume of the given solid. A cone with height \(h\) and base radius \(r\)
46 Theorem of Pappus · Level 3
Use the Theorem of Pappus to find the volume of the given solid. The solid obtained by rotating the triangle with vertices \((2, 3)\), \((2, 5)\), and \((5, 4)\) about the \(x\)-axis
47 Centroid of a Curve · Level 3
The centroid of a curve can be found by a process similar to the one we used for finding the centroid of a region. If \(C\) is a curve with length \(L\), then the centroid is \((\bar{x}, \bar{y})\) where \(\bar{x} = \left(\dfrac{1}{L}\right) \int x d s\) and \(\bar{y} = \left(\dfrac{1}{L}\right) \int y d s\). Here we assign appropriate limits of integration, and \(d s\) is as defined in Sections 8.1 and 8.2. (The centroid often doesn't lie on the curve itself. If the curve were made of wire and placed on a weightless board, the centroid would be the balance point on the board.) Find the centroid of the quarter-circle \(y = \sqrt{16 - x^2}\), \(0 \leq x \leq 4\).
48 Second Theorem of Pappus · Level 4
The Second Theorem of Pappus is in the same spirit as Pappus's Theorem on page 565, but for surface area rather than volume: Let \(C\) be a curve that lies entirely on one side of a line \(l\) in the plane. If \(C\) is rotated about \(l\), then the area of the resulting surface is the product of the arc length of \(C\) and the distance traveled by the centroid of \(C\) (see Exercise 47).
(a) Prove the Second Theorem of Pappus for the case where \(C\) is given by \(y = f(x)\), \(f(x) \geq 0\), and \(C\) is rotated about the \(x\)-axis.
(b) Use the Second Theorem of Pappus to compute the surface area of the half-sphere obtained by rotating the curve from Exercise 47 about the \(x\)-axis. Does your answer agree with the one given by geometric formulas?

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49 Second Theorem of Pappus · Level 3
Use the Second Theorem of Pappus described in Exercise 48 to find the surface area of the torus in Example 7.
50 Centroid of Region · Level 4
Let \(cal(R)\) be the region that lies between the curves \(y = x^m\), \(y = x^n\), \(0 \leq x \leq 1\), where \(m\) and \(n\) are integers with \(0 \leq n < m\).
(a) Sketch the region \(cal(R)\).
(b) Find the coordinates of the centroid of \(cal(R)\).
(c) Try to find values of \(m\) and \(n\) such that the centroid lies outside \(cal(R)\).

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51 Centroid (Proof) · Level 5
Prove Formulas 9.
52 Example - Hydrostatic Force on Dam · Level 3
A dam has the shape of a trapezoid. The height is 20 m and the width is 50 m at the top and 30 m at the bottom. Find the force on the dam due to hydrostatic pressure if the water level is 4 m from the top of the dam.
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53 Example - Hydrostatic Force on Drum · Level 3
Find the hydrostatic force on one end of a cylindrical drum with radius 3 ft if the drum is submerged in water 10 ft deep.
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54 Example - Center of Mass of System · Level 2
Find the moments and center of mass of the system of objects that have masses 3, 4, and 8 at the points \((-1, 1)\), \((2, -1)\), and \((3, 2)\), respectively.
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55 Example - Centroid of Semicircular Plate · Level 3
Find the center of mass of a semicircular plate of radius \(r\).
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56 Example - Centroid of Region · Level 3
Find the centroid of the region bounded by the curves \(y = \cos x\), \(y = 0\), \(x = 0\), and \(x = \dfrac{\pi}{2}\).
57 Example - Centroid of region between curves · Level 2
Find the centroid of the region bounded by the line \(y = x\) and the parabola \(y = x^2\).
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58 Example - Theorem of Pappus · Level 3
A torus is formed by rotating a circle of radius \(r\) about a line in the plane of the circle that is a distance \(R\) \((R > r)\) from the center of the circle. Find the volume of the torus.

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