Find the lengths of the curves in Exercises 1720.

The solid lies between planes perpendicular to the x-axis at and The cross-sections perpendicular to the x-axis between these planes are circular disks whose diameters run from the parabola to the parabola y = x y = 2x.

The base of the solid is the region in the first quadrant between the line and the parabola The cross-sections of the solid perpendicular to the x-axis are equilateral triangles whose bases stretch from the line to the curve.

The solid lies between planes perpendicular to the x-axis at and The cross-sections between these planes are circular

The solid lies between planes perpendicular to the x-axis at and The cross-sections between these planes are squares whose bases run from the x-axis up to the curve

The solid lies between planes perpendicular to the x-axis at and The cross-sections of the solid perpendicular to the x-axis between these planes are circular disks whose diameters run from the curve to the curve

The base of the solid is the region bounded by the parabola and the line in the xy-plane. Each cross-section perpendicular to the x-axis is an equilateral triangle with one edge in the plane. (The triangles all lie on the same side of the plane.)

Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve and the lines and about (a) the x-axis; (b) the y-axis; (c) the line (d) the line

Find the volume of the solid generated by revolving the triangular region bounded by the curve and the lines and about (a) the x-axis; (b) the y-axis; (c) the line (d) the line

Find the volume of the solid generated by revolving the region bounded on the left by the parabola and on the right by the line about (a) the x-axis; (b) the y-axis; (c) the line

Find the volume of the solid generated by revolving the region bounded by the parabola and the line about (a) the x-axis; (b) the y-axis; (c) the line (d) the line

Find the volume of the solid generated by revolving the triangular region bounded by the x-axis, the line and the curve in the first quadrant about the x-axi

Find the volume of the solid generated by revolving the region bounded by the curve and the lines and about the line

Find the volume of the solid generated by revolving the region bounded by the curve and the lines and about the x-axi

Find the volume of the solid generated by revolving about the x-axis the region bounded by and (The region lies in the first and third quadrants and resembles a skewed bowtie.)

A round hole of radius is bored through the center of a solid sphere of a radius 2 ft. Find the volume of material removed from the sphere

The profile of a football resembles the ellipse shown here. Find the footballs volume to the nearest cubic inch.

Find the lengths of the curves in Exercises 1720.

Find the lengths of the curves in Exercises 1720.

Find the lengths of the curves in Exercises 1720.

Find the lengths of the curves in Exercises 1720.

In Exercises 2124, find the areas of the surfaces generated by revolving the curves about the given axes

In Exercises 2124, find the areas of the surfaces generated by revolving the curves about the given axes

In Exercises 2124, find the areas of the surfaces generated by revolving the curves about the given axes

In Exercises 2124, find the areas of the surfaces generated by revolving the curves about the given axes

A rock climber is about to haul up 100 N (about 22.5 lb) of equipment that has been hanging beneath her on 40 m of rope that weighs 0.8 newton per meter. How much work will it take? (Hint: Solve for the rope and equipment separately, then add.)

You drove an 800-gal tank truck of water from the base of Mt. Washington to the summit and discovered on arrival that the tank was only half full. You started with a full tank, climbed at a steady rate, and accomplished the 4750-ft elevation change in 50 min. Assuming that the water leaked out at a steady rate, how much work was spent in carrying water to the top? Do not count the work done in getting yourself and the truck there. Water weighs 8 lb U.S. gal.

You drove an 800-gal tank truck of water from the base of Mt. Washington to the summit and discovered on arrival that the tank was only half full. You started with a full tank, climbed at a steady rate, and accomplished the 4750-ft elevation change in 50 min. Assuming that the water leaked out at a steady rate, how much work was spent in carrying water to the top? Do not count the work done in getting yourself and the truck there. Water weighs 8 lb U.S. gal.

A force of 200 N will stretch a garage door spring 0.8 m beyond its unstressed length. How far will a 300-N force stretch the spring? How much work does it take to stretch the spring this far from its unstressed length?

A reservoir shaped like a right-circular cone, point down, 20 ft across the top and 8 ft deep, is full of water. How much work does it take to pump the water to a level 6 ft above the top?

(Continuation of Exercise 29.) The reservoir is filled to a depth of 5 ft, and the water is to be pumped to the same level as the top. How much work does it take?

A right-circular conical tank, point down, with top radius 5 ft and height 10 ft is filled with a liquid whose weight-density is How much work does it take to pump the liquid to a point 2 ft above the tank? If the pump is driven by a motor rated at 275 ft-lb sec (1 2 hp), how long will it take to empty the tank?

A storage tank is a right-circular cylinder 20 ft long and 8 ft in diameter with its axis horizontal. If the tank is half full of olive oil weighing find the work done in emptying it through a pipe that runs from the bottom of the tank to an outlet that is 6 ft above the top of the tank.

Find the centroid of a thin, flat plate covering the region enclosed by the parabolas

Find the centroid of a thin, flat plate covering the region enclosed by the x-axis, the lines and and the parabola

Find the centroid of a thin, flat plate covering the triangular region in the first quadrant bounded by the y-axis, the parabola and the line

Find the centroid of a thin, flat plate covering the region enclosed by the parabola and the line

Find the center of mass of a thin, flat plate covering the region enclosed by the parabola and the line if the density function is (Use horizontal strips.)

a. Find the center of mass of a thin plate of constant density covering the region between the curve and the x-axis from to b. Find the plates center of mass if, instead of being constant, the density is (Use vertical strips.)

The vertical triangular plate shown here is the end plate of a trough full of water What is the fluid force against the plate?

The vertical trapezoidal plate shown here is the end plate of a trough full of maple syrup weighing What is the force exerted by the syrup against the end plate of the trough when the syrup is 10 in. deep?

A flat vertical gate in the face of a dam is shaped like the parabolic region between the curve and the line with measurements in feet. The top of the gate lies 5 ft below the surface of the water. Find the force exerted by the water against the gate

You plan to store mercury in a vertical rectangular tank with a 1 ft square base side whose interior side wall can withstand a total fluid force of 40,000 lb. About how many cubic feet of mercury can you store in the tank at any one time?