Find the area of the region. $f(x)=x\left(x^{2}-3 x+3\right), \ g(x)=x^{2}$

Find the arc length from (0, 3) clockwise to (2, √5) along the circle x²+y²=9.

Use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis.

$$y=\sqrt{1-x^3}, \quad y=0, \quad x=0$$

(a) use a graphing utility to graph the plane region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis. y = √1 - x³, y=0, x=0

In this exercise, find and/or verify the centroid of the common region used in engineering.

Triangle Show that the centroid of the triangle with vertices $(-a, 0),(a, 0)$, and $(b, c)$ is the point of intersection of the medians (see figure).In this exercise, find and/or verify the centroid of the common region used in engineering.

Triangle Show that the centroid of the triangle with vertices $(-a, 0),(a, 0)$, and $(b, c)$ is the point of intersection of the medians (see figure).

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

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. Verify your results using the integration capabilities of a graphing utility. y = e^(x-1), y=0, x=1, x=2

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

In this exercise, consider a 15-foot hanging chain that weighs 3 pounds per foot. Find the work done in lifting the chain vertically to the indicated position.

Repeat the previous exercise raising the bottom of the chain to the 12-foot level.

The Gateway Arch in St. Louis, Missouri, is modeled by y = 693.8597 - 68.7672 cosh 0.0100333x, -299.2239 ≤ x ≤ 299.2239. Use the integration capabilities of a graphing utility to approximate the length of this curve.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Parabola, y=x², 4 feet across the top and with a height of 4 feet,

Use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis.

$x^{4 / 3}+y^{4 / 3}=1, \quad x=0, \quad y=0$, first quadrant

(a) use a graphing utility to graph the plane region bounded by the graphs of the equations, and (b) use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis. x^4/3 + y^4/3 = 1, x = 0, y = 0, first quadrant.

Explain why fluid pressure on a surface is calculated using horizontal representative rectangles instead of vertical representative rectangles.