?A vertical plate is submerged (or partially submerged) in water and has the indicated

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. Equation Transcription: Text Transcription: 5 ft 2 ft 3 ft

A tank is \(8 \mathrm{~m}\) long, \(4 \mathrm{~m}\) wide, \(2 \mathrm{~m}\) high, and contains kerosene with density \(820 \mathrm{~kg} / \mathrm{m}^{3}\) to a depth of \(1.5 \mathrm{~m}\). 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. Equation Transcription: Text Transcription: 8 m 4 m 2 m 820 kg/m^3 1.5 m Image text transcription: A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density 820 kg/m3 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.

A tanker truck transports gasoline in a horizontal cylindrical tank with diameter \(8 f t\) and length \(40ft\). If the tank is full of gasoline with density \(47 l b / f t^{3}\), compute the force exerted on one end of the tank. Equation Transcription: Text Transcription: 8 ft 40 ft 47 lb /ft^3

A trough with a trapezoidal cross-section, as shown in the figure, contains vegetable oil with density \(925 \mathrm{~kg} / \mathrm{m}^{3}\) (a) Find the hydrostatic force on one end of the trough if it is completely full of oil. (b) Compute the force on one end if the trough is filled to a depth of \(1.2 \mathrm{~m}\). Equation Transcription: Text Transcription: 925 kg/m^3 1.2 m

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.

A dam is inclined at an angle of from the vertical and has the shape of an isosceles trapezoid wide at the top and 50 ft wide at the bottom and with a slant height of . Find the hydrostatic force on the dam when the water level is at the top of the dam.

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) each of the four sides and (b) the bottom of the pool.

Suppose that a plate is immersed vertically in a fluid with density and the width of the plate is at a depth of meters beneath the surface of the fluid. If the top of the plate is at depth and the bottom is at depth , show that the hydrostatic force on one side of the plate is

A metal plate was found submerged vertically in seawater, which has density . Measurements of the width of the plate were taken at the indicated depths. Use the formula in Exercise 18 and Simpson's Rule to estimate the force of the water against the plate. Depth (ft) Plate width (ft)

(a) Use the formula in Exercise 18 to show that \(F=(\rho \bar{x}) A\) 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.

Point-masses are located on the -axis as shown. Find the moment of the system about the origin and the center of mass .

Point-masses are located on the -axis as shown. Find the moment of the system about the origin and the center of mass .

The masses are located at the points . Find the moments and and the center of mass of the system. ;

The masses are located at the points . Find the moments and and the center of mass of the system. ;

Visually estimate the location of the centroid of the region shown. Then find the exact coordinates of the centroid.

Visually estimate the location of the centroid of the region shown. Then find the exact coordinates of the centroid.

Visually estimate the location of the centroid of the region shown. Then find the exact coordinates of the centroid.

Visually estimate the location of the centroid of the region shown. Then find the exact coordinates of the centroid.

Find the centroid of the region bounded by the given curves.

Find the centroid of the region bounded by the given curves.

Find the centroid of the region bounded by the given curves. \(y=\sin 2 x, \quad y=\sin x, \quad 0 \leqslant x \leqslant \pi / 3\) Equation Transcription: y = sin 2x, y = sin x, 0 ? x ? Text Transcription: y = sin 2x, y = sin x, 0 leqslant x leqslant pi/3

Find the centroid of the region bounded by the given curves.

Find the centroid of the region bounded by the given curves.

Calculate the moments and and the center of mass of a lamina with the given density and shape.

Calculate the moments and and the center of mass of a lamina with the given density and shape.

Use Simpson's Rule to estimate the centroid of the region shown.

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.

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. Equation Transcription: Text Transcription: x-coordinate y=e^x y=2-x^2

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.] Equation Transcription: Text Transcription: (a,0), (0,b), (c,0)

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.

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.

A rectangle \(\mathscr{R}\) with sides \(a\) and \(b\) is divided into two parts \(\mathscr{R}_{1}\) and \(\mathscr{R}_{2}\) by an arc of a parabola that has its vertex at one corner of \(\mathscr{R}\) and passes through the opposite corner. Find the centroids of both \(\mathscr{R}_{1}\) and \(\mathscr{R}_{2}\). Equation Transcription: ? ?1 ?2 ? ?1 ?2 Text Transcription: R a b R_1 R_2 R R_1 R_2

If \(\bar{x}\) is the \(x\)-coordinate of the centroid of the region that lies under the graph of a continuous function \(f\), where \(a \leqslant x \leqslant b\), show that \(\int_{a}^{b}(c x+d) f(x) \ d x=(c \bar{x}+d) \int_{a}^{b} f(x) \ d x\) Equation Transcription: ? ? Text Transcription: bar x x-coordinate f a leqslant x leqslant b Integral over a ^b (cx+d)f(x)dx=(c bar x+d)Integral over a ^b f(x)dx

Use the Theorem of Pappus to find the volume of the given solid. A sphere of radius \(r \quad\) (Use Example 4.) Equation Transcription: Text Transcription: r

Use the Theorem of Pappus to find the volume of the given solid. A cone with height \(h\) and base radius \(r\). Equation Transcription: Text Transcription: h r

The solid obtained by rotating the triangle with vertices \((2,3),(2,5)\), and \((5,4)\) about the \(x\)-axis. Equation Transcription: Text Transcription: (2,3), (2,5), (5,4) x-axis

Centroid of a Curve 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 \(\bar{x}=(1 / L) \int x \ d s\) and \(\bar{y}=(1 / L) \int y \ d s\). 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 \(y=\sqrt{16-x^{2}}, 0 \leqslant x \leqslant 4\) Equation Transcription: ? ? Text Transcription: C L (bar x, bar y) bar x=(1/L)x ds bary=(1/L)y ds ds y=sqrt 16-x2, 0 leqslant x leqslant 4

The Second Theorem of Pappus The Second Theorem of Pappus is in the same spirit as Pappus's Theorem discussed in this section, 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 \(y=f(x), f(x) \geqslant 0\), 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? Equation Transcription: ? Text Transcription: C l C l C C y=f(x), f(x) geqslant 0 C x-axis x-axis

The Second Theorem of Pappus The Second Theorem of Pappus is in the same spirit as Pappus's Theorem discussed in this section, 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). Use the Second Theorem of Pappus to find the surface area of the torus in Example 7.

Let \(\mathscr{R}\) be the region that lies between the curves \(y=x^{m} \quad y=x^{n} \quad 0 \leqslant x \leqslant 1\) where \(m\) and \(n\) are integers with \(0 \leqslant n<m\). (a) Sketch the region \(\mathscr{R}\). (b) Find the coordinates of the centroid of \(\mathscr{R}\). (c) Try to find values of \(m\) and \(n\) such that the centroid lies outside \(\mathscr{R}\) Equation Transcription: ? ? ? ? ? ? ? ? Text Transcription: R y=x^m y=x^n 0 leqslant x leqslant 1 m n 0 leqslant n leqslant m R R m n R

Prove Formulas