Area between curves
Volume using Disk method
Volume using Washer method
Volume of Solids around vertical and horizontal lines
Volume of cross sections
100
Find the integral that gives you the area of the region bounded by y=4 and
y=x^2
\int_-2^2 4-x^2dx
100
What is the Disk Method?
A formula for finding the Volume of a solid formed by revolving a function around a line.
V=\int_a ^b\pi(f(x))^2dx
100
What is the washer method?
A formula for finding the Volume of a solid that has a cavity formed by revolving a function around a line.
\int_a^b\pi(R^2(x)-r^2(x))dx
100
Find the volume of a solid of region R, bounded by y=2, y=7,x=1,x=10, revolved around the y=7.
225pi
100
What is the formula for the volume of a 3D shape with base bounded between g(x) and f(x) (g(x)>f(x)) on the interval [a,b] and its cross sections made of equilateral triangles perpendicular to the x-axis?
V=int_a^b (sqrt3)/4(g(x)-f(x))^2dx
200
Find the area of the region R.
4.95
200
Consider a region bounded by the graphs below. Write but do not evaluate the integral that would give you the volume of the solid formed by revolving this region around the x-axis.
y=ln x
y=0
x=e
int_1^e pi(lnx)^2dx
200
Determine the volume of the solid obtained by rotating the region bounded by y=√x, y=3 and the y-axis about the x-axis.
(81pi)/2
200
Find the volume of the solid generated when 𝑇 is revolved about the vertical line 𝑥 =6.
407.15
200
Let R be a region bounded by the graphs below. Find the volume of a solid that has R as its base with every cross section being a square perpendicular to the x-axis.
x=y^2
x=9
162
400
Find the area of R. If R is the region between the graphs cos(x) and sin(x) over the interval
[0,\pi]
2sqrt2
400
Determine the volume of the solid obtained by rotating the region bounded by y=√x, y=3 and the y-axis about the y-axis.
243/5pi
400
(Calculator)Determine the volume of a solid generated by rotating the region R around the x-axis. The region R is bounded by
y=-x^2+4
y=x^2+2
16pi
400
Find the volume of a solid revolved around y=2.
37.44
400
Let R be a region bounded by the graphs below. Find the volume of a solid that has R as its base with every cross section being a triangle with height equal to 1/4 the length of the base perpendicular to the x-axis.
x=y^2
x=9
81/4
500
Find the area of the region R (below) by integrating with respect to y.
5/3
500
Consider a region bounded by the graphs below. Find the volume of the solid formed by revolving this region around the y-axis.
2x+3y=6
y=0
x=0
6pi
500
Find the volume of a sphere with radius 4 that is hollowed by another sphere of radius 2 using integrals.
234.57
500
(Calculator)Determine the volume of a solid generated by rotating the region R around the x=-3. The region R is bounded by
y=e^x
y=1
x=ln3
18.59pi
500
Suppose you have a solid with an elliptical base whose equation is given below, semicircular cross-sectional slices are taken perpendicular to the y-axis. Find the volume of the solid.
x^2+4y^2=1
pi/3