Average Value, MVT
This is the average value of f(x)=3x+2 on the interval 0<=x<=1.
What is 1?
This is the area of the region enclosed by the graphs of y=1+x^2, y=1-x, and x=1.
What is 5/6?
Let R be the region bounded by the graphs of y=1-x^2, the x-axis, and the y-axis. This is the volume of the solid obtained by revolving R about the x-axis.
What is (8pi)/15?
Let R be the region bounded by the graphs of y=sqrt(x), x=3, and the x-axis. This is the volume of the solid whose base is R and whose cross-sections perpendicular to the x-axis are squares.
What is 9/2?
This is the formula for the arc length traversed by a smooth function f(x) on the interval a<=x<=b.
What is int_a^b sqrt(1+(f’(x))^2)dx?
This is the average value of f(x)=-1/x^2 on the interval 2<=x<=3.
What is -1/6?
This is the area of the region enclosed by the graphs of y=e^x, y=e^(-x), and x=ln(100) expressed as a single fraction in simplified form.
What is 9801 / 100?
Let x be the region bounded by the graphs of y=1-x^2, the x-axis and the y-axis. This is the volume of the solid obtained by revolving R about the y-axis.
What is pi/2?
Let R be the region bounded by the graphs of y=x and y=x^2. This is the volume of the solid whose base is R and whose cross-sections perpendicular to the x-axis are isosceles right triangles with their right angles along the x-axis.
What is 1/60?
This is the arc length traversed by f(x)= -sqrt(5)/2x on 0<=x<=1 expressed as a fraction in simplest form.
What is 3/2?
For f(x)=2x on the interval 0<=x<=1, this is the value of c which is guaranteed to exist by the Mean Value Theorem for Integrals.
What is c=1/2?
This is the area of the region enclosed by the graphs of x=y^2 and y=x/2 expressed as a single fraction in simplified form.
What is 4/3?
Let R be the region bounded by the graphs of y=sqrt(x), x=4, and the x-axis. This is the volume of the solid obtained by revolving R about the line y=-3.
What is 76pi?
Let R be the region bounded by the graphs of y=x^2 and y=1. This is the volume of the solid whose base is R and whose cross-sections perpendicular to the y-axis are quarter circles with their radius spanning the region R.
What is pi/2?
This is the arc length traversed by f(x)= sqrt(21)/3 * x^(3/2) on 0<=x<=1, expressed as a fraction in simplest form.
What is 13/7?
This is the average value of f(x)=4sqrt(x) on the interval 0<=x<=3.
What is (8sqrt(3))/3?
This is the area of the region enclosed by the graphs of y=x, y=1/x^2, and the x -axis. 200 point bonus if you integrate with respect to y.
What is 3/2?
Let R be the region in the first quadrant bounded by the graphs of y=x^3 and y=x^(⅓). This is the volume of the solid obtained by revolving R about the y-axis.
What is (16pi)/35?
Let R be the region bounded by the graphs of y=ln(x), y=0, and y=1. This is the volume of the solid whose base is R and whose cross-sections perpendicular to the y-axis are semicircles.
What is pi/16?
This is the arc length traversed by f(x)= x^3/6 + 1/(2x) on 1<=x<=3 expressed as a fraction in simplest form.
What is 14/3?
For f(x)=4/(2x+6)^2 on the interval -6<=x<=5, this is the value of c which is guaranteed to exist by the Mean Value Theorem for Integrals.
What is -3 - sqrt(6)?
This is the area of the region enclosed by the graphs of y=2xsin(x^2) and y=2xcos(x^2).
What is sqrt(2)?
Let R be the region bounded by the graphs of y=3x-2 and y=x^2. This is the volume of the solid obtained by revolving R about the line x=-1.
What is (5pi)/6?
(CALCULATORS ALLOWED): Let R be the region bounded by the graphs of y=3x and y=x^2 + 2. This is the volume of the solid whose base is R and whose cross-sections perpendicular to the y-axis are equilateral triangles. Give your answer either in exact form or rounded to at least 3 decimal places.
What is 3531*sqrt(3)/20, or ~305.794?
This is the arc length traversed by f(x)=sqrt(4-x^2) on -1<=x<=1.
What is (2pi)/3?