Exam 8: Applications of Integration

Use Pappus's Theorem to find the volume of the solid of revolution obtained by rotating the triangular plane region specified by 0 $\le$ y $\le$ 1 - about (a) the line x = 2 and(b) the line y = 2.

Find the centre of mass of the semicircular plate 0 $\le$ y $\le$ assuming it has constant density.

For any events $\textbf{ A }$ and $\textbf{ B }$ in a sample space $\textbf{ S, Pr }$ ( $\textbf{ A }$ or $\textbf{ B }$ ) = $\textbf{ Pr(A) + Pr(B) + Pr(A and B). }$

Find the volume of a solid formed by revolving the disk bounded by a circle of radius a cm about a line tangent to that circle.

Find the length of the arc y = ln(sec x) between x = 0 and x = .

Find the solution of the initial-value problem = , x(0) = a.

Assuming the Earth is spherical with radius 6378 km, find the area of the surface of the Earth between the Tropic of Cancer (23.5° north latitude) and the Antarctic Circle (66.5° south latitude) as shown in the figure below.

Let A(x) be the cross-sectional area of a solid by planes perpendicular to the x-axis. If the volume of the solid that lies between x = 1 and x = z > 1 is V = 4 + 2, find A(x).

The region R is the portion of the first quadrant that is below the parabola y² = 8x and above the hyperbola y² - x² = 15. Find the volume of the solid obtained by revolving R about the x-axis.

The plane region bounded by the curve + = 1 is revolved about the line x = 2. Find the volume of the solid generated.

Consider the finite plane region bounded by the coordinate axes and the line 4x + 3y = 12. Assuming the region has constant areal density 1, find the moments of this region about the coordinate axes.

Find the volume of a solid whose base is the region in the first quadrant bounded by the line and the coordinate axes if every planar section perpendicular to the x-axis is a semicircle.

Determine the solution of the differential equation that satisfies the boundary conditions (x) = 8, , .

Solve the differential equation = 6x .

Let R be the plane region enclosed by the graphs of y = f(x) and y = g(x) from x = a to x = b , where a > 0(as shown in the figure below).If the solid generated by revolving the plane region R about the x-axis has the same volume as the solid generated by revolving the region R about the y-axis , then f and g satisfy which equation for all x > 0?

Find the mass of a thin plate that occupies the planar region described by 0 $\le$ y $\le$ sin(2x), 0 $\le$ x $\le$ if the areal density is given by (x) = 8x.

Find the volume of an elliptical cone whose base in the horizontal xy-plane is the elliptic disk (where a > 0 and b > 0) and whose vertex is at height h directly above the centre of the base.

The force of gravity (outside the Earth) attracting a mass m is proportional to the mass m and inversely proportional to the square of the distance from the centre of the Earth. Find the work done in moving a mass that weighs 1 lb at the surface of the Earth to 10 miles above the surface. Assume the radius of the Earth is 4,000 mi. At the Earth's surface, a force of 1 lb is required to lift a mass of 1 lb.

Find the arc length of the curve x = (y) from y = to y = .

A notch is cut out of a vertical cylindrical log of radius r cm by two planar saw cuts that meet along a horizontal line passing through the centre of the log. If the saw cuts make angles ± 30º with the horizontal (so that the angle of the notch is 60º), find the volume of wood cut out of the log in making the notch.