Answer: A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined

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.

A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density 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.

311 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.

311 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.

311 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.

311 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.

311 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.

311 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.

311 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.

311 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.

311 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 milk truck carries milk with density 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?

A trough is filled with a liquid of density 840 kgm . 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.

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

A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Estimate 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 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

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, estimate the hydro static force on (a) the shallow end, (b) the deep end, (c) one of the sides, and (d) 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 F y b a txwx dx 6

. A metal plate was found submerged vertically in sea water, which has density . Measurements of the width of the plate were taken at the indicated depths. Use Simpsons Rule to estimate the force of the water against the plate.

(a) Use the formula of Exercise 18 to show that where is the -coordinate of the centroid of the plate and 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.

2122 Point-masses are located on the -axis as shown. Find the moment of the system about the origin and the center of mass .0 30 x 10 m=6 m=9

2122 Point-masses are located on the -axis as shown. Find the moment of the system about the origin and the center of mass .x m=15 m=20 _3 m

2324 The masses are located at the points . Find the moments and and the center of mass of the system.m1 4 m2 2 m3 4 P12, 3 P23, 1 P33, 5 mi Pi

2324 The masses are located at the points . Find the moments and and the center of mass of the system.m1 5 m2 4 m3 3 m4 6 P14, 2 P20, 5 P33, 2 P41, 2 y 2x y

2528 Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordi nates of the centroid.y 2x y 0 x 1 y

2528 Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordi nates of the centroid.y sx y 0 x 4 y

2528 Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordi nates of the centroid.y e y 0 x 0 x 1 x y s

2528 Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordi nates of the centroid.y sin x y 0 0 x x

2933 Find the centroid of the region bounded by the given curves.x y 2 y x 2 y

2933 Find the centroid of the region bounded by the given curves.y 2 x y x 2 y

2933 Find the centroid of the region bounded by the given curves.y sin x y cos x x 0 x 4 y x

2933 Find the centroid of the region bounded by the given curves.y x x  y 2 y 0 3 x

2933 Find the centroid of the region bounded by the given curves.x y 2 x  y 2 Mx

34 35 Calculate the moments and and the center of mass of a lamina with the given density and shape. 3 

35 35 Calculate the moments and and the center of mass of a lamina with the given density and shape. 10 x

Use Simpsons Rule to estimate the centroid of the region shown.

Find the centroid of the region bounded by the curves and . Sketch the region and plot the centroid to see if your answer is reasonable.

Use a graph to find approximate -coordinates of the points of intersection of the curves and . Then find (approximately) the centroid of the region bounded by these curves.

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 , , and . 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 twothirds of the way from each vertex (along the median) to the opposite side.]

40 41 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.

40 41 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 with sides and is divided into two parts and by an arc of a parabola that has its vertex at one corner of and passes through the opposite corner. Find the centroids of both and .

If is the -coordinate of the centroid of the region that lies under the graph of a continuous function , where , show that y b a cx  d fx dx cx  dy b a fx dx xx

44 46 Use the Theorem of Pappus to find the volume of the given solid.A sphere of radius (Use Example 4.)

45 46 Use the Theorem of Pappus to find the volume of the given solid.A cone with height and base radius

46 46 Use the Theorem of Pappus to find the volume of the given solid.The solid obtained by rotating the triangle with vertices , , and about the -axis

Prove Formulas 9.

Let be the region that lies between the curves and , , where and are integers with . (a) Sketch the region . (b) Find the coordinates of the centroid of . (c) Try to find values of and such that the centroid lies outside .