Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. $y=e^{-3x}$, y=0, x=0, x=2

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. $y=\frac{x^{3}}{18}$, $3 \leq x \leq 6$

Use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. y = 5∛400-x², y = 0

(a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the arc length. x = e^(-y), 0 ≤ y ≤ 2

Find M_x, M_y, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. x = y + 2, x = y²

Decide whether it is more convenient to use the disk method or the shell method to find the volume of the solid of revolution. Explain your reasoning. y=4-e^x

g(x) = 4 / (2-x), y=4, x=0

Find the work done in pumping gasoline that weighs 42 pounds per cubic foot. The top of a cylindrical gasoline storage tank at a service station is 4 feet below ground level. The axis of the tank is horizontal and its diameter and length are 5 feet and 12 feet, respectively. Find the work done in pumping the entire contents of the full tank to a height of 3 feet above ground level.

(a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the arc length. y = ln x, 1 ≤ x ≤ 5

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y=2/(x+1), y=0, x=0, x=6

Find the arc length of the graph of the function over the indicated interval. $y=\frac{1}{3}x^{3 / 2}-1$, [2, 6]

Two identical semicircular windows are placed at the same depth in the vertical wall of an aquarium. Which is subjected to the greater fluid force? Explain.

Find the work done in pumping gasoline that weighs 42 pounds per cubic foot. (Hint:Evaluate one integral by a geometric formula and the other by observing that the integrand is an odd function.) -A cylindrical gasoline tank 3 feet in diameter and 4 feet long is carried on the back of a truck and is used to fuel tractors. The axis of the tank is horizontal. The opening on the tractor tank is 5 feet above the top of the tank in the truck. Find the work done in pumping the entire contents of the fuel tank into the tractor.

Decide whether it is more convenient to use the disk method or the shell method to find the volume of the solid of revolution. Explain your reasoning. (Do not find the volume.) (y-2)² = 4-x