unit 8 b/c

The regions bounded by the graphs y=x/2 and y=sin⁡2x are shaded in the figure above. What is the sum of the areas of the shaded regions?

0.249

What is the area of the region in the first quadrant bounded on the left by the graph of x=y√y4+1 and on the right by the graph of x=2y ?

0.537

The base of a solid is the region in the first quadrant bounded by the graph of y=sin⁡x and the x-axis for 0≤x≤π. For the solid, each cross section perpendicular to the x-axis is an equilateral triangle. What is the volume of the solid?

0.680

For time t≥0t≥0, the acceleration of an object moving in a straight line is given by a(t)=sin(t23)). What is the net change in velocity from time t=0.75 to time t=2.25 ?

0.984

Let R be the region in the first quadrant, bounded by the graph of y=√x, the y-axis, and the line y=1. The region R is the base of a solid. For the solid, each cross section perpendicular to the y-axis is an isosceles right triangle with the right angle on the y-axis and one leg in the xy-plane. What is the volume of the solid?

1/10

A traffic engineer developed the continuous function R, graphed above, to model the rate at which vehicles pass a certain intersection over an 8-hour time period, where R(t) is measured in vehicles per hour and t is the number of hours after 6:00 AM. According to the model, how many vehicles pass the intersection between time t=0 and time t=8 ?

14,400

The graph of the continuous function f consists of three line segments, as shown in the figure above. What is the average value of f on the interval [−1,6] ?

15/7

Let R be the region bounded by the graphs of y=1/2x2+2 and y=2+6x−x^2, as shown in the figure above. What is the perimeter of region R ?

20.520

The base of a solid is the triangular region in the first quadrant bounded by the graph of y=5−5/3x and the x- and y-axes. For the solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?

25

A particle moves along the xx-axis with velocity given by v(t)=(t−1)e^1−t for time t≥0t≥0. If the particle is at position x=3 at time t=0 which of the following gives the position of the particle at time t=2

3 + ∫20(t−1)e^1−t dt

The base of a solid is the region in the first quadrant bounded by the x- and y-axes and the graph of y=(x−2)^2/2(x+1). For the solid, each cross section perpendicular to the x-axis is a rectangle whose height is three times its width in the xy-plane. What is the volume of the solid?

3.012

What is the volume of the solid generated when the region in the first quadrant bounded by the graph of y=x^3, the y-axis, and the horizontal line y=1 is revolved about the y-axis?

3π/5

Let R be the region in the first quadrant bounded by the graphs of x=y^3 and x=4y, as shown in the figure above. What is the area of R ?

4

What is the volume of the solid generated when the region in the first quadrant bounded by the graph of y=4√4−x^2 and the x- and y-axes is revolved about the y-axis?

67.021

Let R be the region in the first quadrant bounded by the graph of y=2cos⁡(x3), the line y=x4, and the y-axis. What is the volume of the solid generated when R is revolved about the line y=4 ?

74.255

The rate at which sand is poured into a bag is modeled by the function r given by r(t)=15t−2t2, where r(t) is measured in milliliters per second and t is measured in seconds after the sand begins pouring. How many milliliters of sand accumulate in the bag from time t=0 to time t=2 ?

74/3

Let R be the region in the first quadrant bounded by the graphs of y=1−x3 and y=1−x, as shown in the figure above. The region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid?

8/105

Let M be the region in the first quadrant bounded by the graph of y=2x+3, the graph of y=x+3, and the vertical line x=4. What is the volume of the solid generated when region M is revolved about the horizontal line y=2 ?

80π

If the average value of the function ff on the interval 1≤x≤41≤x≤4 is 8, what is the value of ∫^4,1 (3f(x)+2x)dx ?

87

Over the time interval 0≤t≤8, a particle moves along the x-axis. The graph of the particle's velocity at time t, v(t), is shown in the figure above. Over the time interval 0≤t≤8, the particle's displacement is 100 units to the right and the particle travels a total distance of 1875. What is the total distance that the particle travels while moving to the left?

887.5

What is the volume of the solid generated when the region in the first quadrant bounded by the graph of y=x4, the x-axis, and the vertical line x=1 is revolved about the x-axis?

pi/9

Let R be the region in the first quadrant, bounded by the graph of y=2sec⁡x, the line x=π4, and the x- and y-axes. R is the base of a solid whose cross sections perpendicular to the x-axis are semicircles. What is the volume of the solid?

π/2

Let R be the region bounded by the graph of y=√x−1, the horizontal line y=2, and the vertical line x=1, as shown in the figure above. Which of the following gives the volume of the solid generated when the region R is revolved about the vertical line x=1 ?

π∫2, 0, ((y2+1)−1)2ⅆy

Let R be the region in the first quadrant bounded on the right by the graph of y=32x−3, above by the horizontal line y=3, and on the left by the vertical line x=2. Which of the following gives the volume of the solid generated when region R is revolved about the vertical line x=2 ?

π∫3, 0, ((2y+63)−2)2ⅆy

Let S be the region in the first quadrant bounded by the graphs of y=x^2/4⁢ and y=x, as shown in the figure above. The graphs intersect at x=0 and x=4. Which of the following gives the volume of the solid generated when S is revolved about the line y=−2 ?

π∫4, 0, ((x+2)^2−(x^2/4+2)^2) dx

Let R be the region in the first quadrant enclosed by the graphs of y=10e−xsin⁡x and the horizontal line y=2, as shown in the figure above. The xx-coordinates of the two points of intersection of the graphs are x1 and x2, where 0<x1<x2. Which of the following gives the volume of the solid generated when region R is revolved about the horizontal line y=2 ?

π∫x2, x1, (10e−xsinx−2)^2dx

Let f be the function given by f(x)=(x2+x)cos(5x). What is the average value of ff on the closed interval 2≤x≤6

−1.848

The regions bounded by the graphs of y=−2πx+1 and y=cos⁡x are shaded in the figure above. Which of the following gives the sum of the areas of the shaded regions?

∫ π/2 0, (cosx−(−2πx+1))ⅆx + ∫ π, π/2 ((−2πx+1)−cosx)ⅆx

Which of the following gives the length of the curve y=x^2+2x from x=0 to x=5 ?

∫0, 5, √1+(2x+2)^2ⅆx

The regions bounded by the graphs of y=2x and y=3x^2−x^3 are shaded in the figure above. The graphs intersect at x=0, x=1, and x=2. Which of the following gives the sum of the areas of the shaded regions?

∫1, 0, (2x−(3x2−x3))ⅆx + ∫2, 1, ((3x2−x3)−2x)ⅆx

Which of the following expressions gives the length of the curve y=e3x from x=−1 to x=2 ?

∫2, 1, −121+9e6xⅆx

Let R be the region between the graph of y=tan−1⁡x, the x-axis, and the line x=1.5 Which of the following gives the area of region R ?

∫tan−1(1.5) 0, (1.5−tany)Dy

The velocity of a particle moving along the x-axis is given by v(t)=−sin⁡(t−π4). Which of the following gives the total distance traveled by the particle over the time interval 0≤t≤π ?

∫v(t) dt −∫v(t) dt