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Integrals
Differential Equations
Area and Volume
100
If: [ x -- 2 - 5 - 7 - 8 ] [ f(x) -- 10 - 30 - 40 - 20 ] The function f is continuous on the closed interval [2,8] and has values that are given in the table above. Using the sub-intervals [2,5], [5,7], and [7,8], what is the trapezoidal approximation of the "∫ of f(x) from 2 to 8 in terms of x"?
What is 160
100
At any time t ≥0, in days, the rate of growth of a bacteria population is given by y’=ky, where y is the number of bacteria present and k is a constant. The initial population is 1,500 and the population is quadrupled during the first 2 days. By what factor will the population have increased during the first 3 days?
What is 8
100
If the region bounded by the curve f(x)=sec x, the x-axis, y-axis, and the line x=∏/4, is revolved about the x-axis, what is the volume of the resulting solid?
What is π
200
get the integral of (144+144x+36x^2)^1/2 dx
What is 36 (x^2 + 4x + 4)
200
The function y satisfies a differential equation of the form y' = ky for some number k. If you are told that when t = 3 that y is 3 and the rate of change of y is 5 then what is k?
What is 5/3
200
The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x+2y=8. If the cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?
What is 16.755?
300
∫ from 0 to 2 of √((x^2)-4x+4) (square root is over all)
What is -2
300
Consider the differential equation: dy/dx = (3-x) / y let y = f(x) be the particular solution to the given differential equation for 1 < x < 5 such that the line y = -2 is tangent to the graph of f. Find the x-coordinate of the point of tangency, and determine whether f has a local maximum, local minimum, or neither at this point.
What is Radius r = 2. If a tangent line is horizontal
300
Let R be the region in the first quadrant enclosed by the graphs of y = 2x and y = x2. Find the area of R
What is 4/3 square units?
400
integration of (3x+y)^-2 dy ? (You must integrate in terms of y)
What is 9(x^2)y + y^3/3 + (6xy^2)/2 + C
400
Consider the differential equation: dy/dx = (3-x) / y Let y = g(x) be the particular solution to the given differential equation for -2 < x < 8, with the initial condition g(6) = -4. Find y = g(x).
What is r
400
Find the area of the region bounded by the curves. y = cos x, y = sin(2x), x = 0, x = pi/2
What is 1/2?
500
Water is pumped out of a holding tank at a rate of 5 - 5e^-0.15t liters/minute, where t is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later? The answer has to be rounded to one decimal place. The amount of water in the tank after one hour = what??
What is 733.329
500
Consider the differential equation dy/dx = (5x^2) - [6/(y-2)] for y ≠ 2. Let f(x) be the particular solution to this differential equation with the initial condition f(-1) = 4. Is it possible for the x-axis to be tangent to the graph of f at some point?
What is Never
500
Let S be the solid described as follows: The base of S is the region in the xy plane bounded by the curves y=ex,y=2 ,x=2 ,x=3 , The cross-sections of S with a plane through the point x and perpendicular to the x axis are square. Find the area A(x) of a cross-section of S with a plane through a point x and perpendicular to the x axis. Answer (formula) A(x)= ____________
What is (ex - 2)²?