Cross Sections
Particle Motion
Volume
Area
1st, 2nd, and 3rd derivative
100
The space is bounded by y=x^2, the line y=4, and the y-axis. A 3 dimensional solid is made by cross sections perpendicular to the x-axis. what will the volume of the shape be if the cross sections are squares?
256/15 units^3
100
A particle's velocity is known at any time, t greater than or equal to 0, using the function v(t) = (x^3) -x+5. Assume the function is continuous. If the particle's position is 10 units at 0 seconds, find the particle's position from 0 to 5 seconds.
203.75 units
100
The equation y = x^3 is bounded by y=2x and x-axis and is revolved about the y-axis. Find the volume.
13.380 units^3
100
The equation y= 5x+2 is bounded by x = 6 and the y axis. What is the area generated with respect to x?
1.5 units squared
100
If f(x)=x^3-x^2+x, what is f'(x)?
3x^2-2x+1
200
The space bounded by y=x^2, the line x=3, and the x-axis. Suppose a three dimensional solid is made by making cross sections perpendicular to the x-axis. What will the volume of the shape be of the cross sections are equilateral triangles?
(243√3)/20
200
A particle moves along the x-axis with its velocity given by v(t)=2t^3. what is the displacement of the particle from time t=2 to t=4?
120
200
The region enclosed by the y-axis, the line y=2, and the curve y=∛x is revolved about the y-axis. What is the volume of the solid generated?
128π/7
200
What is the area of the region between the graphs of y=x^3 and y=-x-1 from x=0 to x=2?
8
200
If f(x) = 5e^(5x), determine, using interval notation, when f(x) is decreasing and increasing.
Decreasing: ERROR, No Intervals Exist
Increasing:(-infinity, +infinity)
300
A space is bounded by y=x^2 and y=x. Assume that a 3 dimensional solid is made by making cross sections. What will the volume of the shape be if the cross sections are semi-circles?
π/240
300
A particle's velocity is determined at any time , t greater than or equal to zero, using the function f(x)=9.8(x)+5 . When, during the first 5 seconds, will the average velocity equal two times the instantaneous velocity?
.995 seconds
300
The Yellow Region is bounded by y=(x-1)^(1/2) and y= (x-1)^(2) and is revolved about the x-axis. Find the Volume of the shape generated.
.943 units^3
300
The equation y = (16-x^2)^1/2 is bounded by y = (-3/2)x+4 and y = (1/2)x . What is the area of the shape generated with respect to x?
4.857 units squared
300
If f(x)= ((x^3)/3)+5((x^2)/2)+6x, determine when f(x) is both increasing and concave up. Also find when f(x) is both decreasing and concave down.
f(x) is concave up and increasing at (-2,+infinity)
f(x) is concave down and decreasing at (-3,-2.5)
400
The base of a solid is the region enclosed by the graph of y=3(x-2)^(2) and the coordinate axes. If every cross section perpendicular to the x-axis is a square, then what is the volume of the solid generated?
57.6 units ^3
400
A sugar ant crawls along the vertical edge of a cereal box with a velocity given by v(t) = 2(-t) + (t-2)^(3) - (t-3)^(2) , for the interval [0,5]. For what intervals is the speed of the sugar ant decreasing?
(-infinity,2)u(8/3,4.116)
400
Let C be the region in the first quadrant enclosed by the lines x= 0 and y = 2 and the graph of y = e^(x). What is the volume of the solid generated when C is revolved about the x-axis?
3.998 units^3
400
An ice field is melting at the rate M(t) = 4-(sin(t))^3 acre-feet per day, where t is measured in days. How many acre-feet of this ice field will melt from beginning of day 1 (t=0) to the beginning of day 4 (t=3).
10.667
400
find the derivative of lnx
1/x