Exam 8: Applications of Integration

Let R be the plane region enclosed by the trapezoid with vertices at the points (0, 0), (1, 1), (5, 1), and (6, 0).(i) Find the y-coordinate of the centroid of the region R.(ii) Use Pappus's Theorem to find the volume of the solid generated by revolving the region R about the x-axis.

Solve the following initial-value problem: = 9.8 - 0.196v,v(0) = 48.

The price of a computer chip t months after it is introduced to the market is given by $C where and the number sold per month at that time is N = 6,000 + 10t.To the nearest thousand dollars, how much income is derived from the sales of these chips over the first year?

Solve the following initial-value problem:

Find the volume of the solid ring obtained by rotating the disc x² + = 9 about the x-axis.

A cube has edge length a cm and one corner at position O. A plane passing through the three corners of the cube that are adjacent to corner O slices the cube into two pieces. Find the volume of the smaller piece.

Solve the initial-value problem = - , y(0) = 4.

Find the moment about the x-axis of a plate of constant areal density 1 occupying the finite plane region bounded by the x-axis and the curve y = -16 + 10x - .

Find the area of the oval surface obtained by rotating the ellipse + 4 = 1 about its major axis (i.e., about the x-axis).

Find the volume of the solid generated by revolving the plane region bounded by the graphs of and the line y = 3 from x = 0 to x = 2ln(2) about the x-axis.

Find the length of the arc y = ln between x = 1 and x = 2.

Find the mean μ and the standard deviation σ of a random variable X distributed uniformly on the interval [2 - 4 , 2 + 4 ].

Find the length of the curve y = - from x = 0 to x = 3.

Determine a multiplier (integrating factor) of the first order linear differential equation+ = x.

Find the solution of the initial-value problem = , x(0) = 5.

Find the area of the surface generated by rotating y = - where x [0, 3] about y = 0.

A finite region R is contained in the first quadrant of the xy-plane. The centroid of R is the point (3, h). When R is revolved about the y-axis it generates a solid having volume 12 cubic units. When R is revolved about the x-axis it generates a solid having volume 32 cubic units. Find (a) the area of R and (b) the value of h.

A ball of radius r has volume V(r) = $\pi$ cubic units. This volume can be regarded as a sum of volumes of concentric spherical shells having radii x units (where 0 $\le$ x $\le$ r) and thickness dx. Use this fact to find the surface area S(r) of a sphere of radius r.

Find the area of the surface generated by rotating + = about y = a.

Solve the initial-value problem + x = , x(2) = -1.