Area In Problems 1–5, find the area \(A\) enclosed by the graphs of the given equations.
\(y=e^{x}, x=0, y=4\)
\(y=x^{2}, y=18-x^{2}\)
\(x=2y^{2}, x=2\)
\(y=\dfrac{1}{x},\) \(x+\) \(y=4\)
\(y=4-x^{2},\) \(y=3x\)
Volume of a Solid of Revolution In Problems 6–15, find the volume of the solid of revolution generated by revolving the region bounded by the graphs of the given equations about the given line.
\(y=x^{2},\) \(\ y=4x-x^{2}\); about the \(x\)-axis
\(y=x^{2}-5x+6, y=0\); about the \(y\)-axis
\(x=y^{2}-4, \ x=0\); about the \(y\)-axis
\(xy=1\), \(x=1\), \(x=2\), \(y=0\); about the \(x\)-axis
\(y=x^{2}-4,\) \(y=0;\) about the line \(y=-4\)
\(y=4x-x^{2}\) and the \(x\)-axis; about the \(x\)-axis
\(y^{2}=8x\), \(y\geq 0\), and \(x=2;\) about the line \(x=2\)
\(y=\dfrac{x^{3}}{2}\), \(y=0\), \(x=2;\) about the \(y\)-axis
\(y=e^{x}\), \(y=1\), \(x=1;\) about the \(x\)-axis
\(y^{2}=x^{3}\), \(y=8\), \(x=0\); about the line \(x=4\)
Arc Length In Problems 16–18, find the arc length of each graph.
\(y=x^{3/2}+4\) from \(x=2\) to \(x=5\)
\(y=\dfrac{x^{3}}{6}+\dfrac{1}{2x}\) from \(x=2\) to \(x=6\)
\(2y^{3}=x^{2}\) from \(y=0\) to \(y=2\)
Center of Mass Find the center of mass of the system of masses: \(m_{1}=1, m_{2}=3, m_{3}=8, m_{4}=1\) located, respectively, at \(-1, 2, 14, \textrm{ and } 0\).
Moments and Center of Mass Find the moments \(M_{x}\) and \(M_{y}\) and the center of mass of the system of masses: \( m_{1}=2, m_{2}=2, m_{3}=3, m_{4}=2\) located, respectively, at the points \( (-4,4), (2,3), (4,4), (-3,-5)\).
Area of a Triangle Use integration to find the area of the triangle formed by the lines \(x-y+1=0\), \(3x+y-13=0\), and \(x+3y-7=0\).
Volume A solid has a circular base of radius \(4\) units. Slices taken perpendicular to a fixed diameter are equilateral triangles. Find its volume.
Volume Find the volume of the solid generated by revolving the region bounded by the graph of \(4x^2+9y^2=36\) in the first quadrant about the \(x\)-axis.
Volume of a Cone Find the volume of an elliptical cone with base \(\dfrac{x^{2}}{4}+y^{2}=1\) and height \(5\). {Hint: The area \(A\) of an ellipse with semi-axes \(a\) and \(b\) is \(A=\pi ab\).)
Volume The base of a solid is enclosed by \(4x+5y=20\), \(x=0\), \(y=0\). Every cross section perpendicular to the base along the \(x\)-axis is a semicircle.Find the volume of the solid.
Arc Length Find the point \(P\) on \(y=\dfrac{2}{3}x^{3/2}\) to the right of the \(y\)-axis so that the length of the curve from \((0,0)\) to \(P\) is \(\dfrac{52}{3}\).
Work Find the work done in raising an 800-\(\textrm{ lb}\) anchor \(150\textrm{ ft}\) with a chain weighing \(20\textrm{ lb/ft}\).
Work Find the work done in raising a container of \(1000\textrm{ kg}\) of silver ore from a mine \(1200\textrm{ m}\) deep with a cable weighing \(3\textrm{ kg/m}\).
Work Pumping Water A hemispherical water tank has a diameter of \(12{m}\). It is filled to a depth of \(4{m}\). How much work is done in pumping all the water over the edge? (Use \(\rho =1000\textrm{ kg/m}^{3}\).)
Work of a Spring A spring with an unstretched length of \(0.6 \textrm{ m}\) requires a force of \(4\textrm{ N}\) to stretch it to \(0.8\textrm{ m}\). How much work is done in stretching it to \(1.4\textrm{ m}\)?
Work of a Spring Find the unstretched length of a spring if the work required to stretch the spring from \(1.0\) to \(1.4\textrm{ m}\) is half the work required to stretch it from \(1.2\) to \(1.8{\text{m}}\).
Hydrostatic Force A trough of trapezoidal cross section is \(2\) ft wide at the bottom, \(4\) ft wide at the top, and \(3\) ft deep. What is the force due to liquid pressure on the end, if it is full of water?
Hydrostatic Force A cylindrical tank is on its side. It has a diameter of \(10{m}\) and is full to a depth of \(5\text{m}\) with gasoline that has a density of \(737\text{kg/m}^{3}\). What is the force due to liquid pressure on the end?
Hydrostatic Force A dam is built in the shape of a trapezoid \(1000{ft}\) long at the top, \(700{ft}\) long at the bottom, and \(80\textrm{ ft}\) deep. Determine the force of water on the dam if:
Centroid Find the centroid of the lamina bounded by the graphs of \(y=\sqrt{x}, y=0\), and \(x=9\).
Pappus Theorem for Volume Find the volume of the torus with an outer diameter of \(5\text{cm}\) and an inner diameter of \(2\text{cm}\).
Volume of a Solid of Revolution Find the volume generated when the triangular region bounded by the lines \(x=3, y=0\), and \(2x+y-12=0\) is revolved about the \(y\)-axis.
Volume of a Solid of Revolution Find the volume generated when the region bounded by the graphs of \(y=3 \sqrt{x}\) and \(y=-x^{2}+6x-2\) is revolved about each of the following lines: