Unit 8 Progress Check: MCQ Part A - AP Calculus BC (2022-2023)

What is the area of the region in the first quadrant bounded on the left by the graph of x=y(y^4 +1)^(1/2) and on the right by the graph of x=2y?

Answer A: 0.537

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?

Answer A: 4

Let R be the region in the first quadrant bounded by the graphs of y=1−x^3 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?

Answer A: 8/105

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

Answer B: -1.848

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?

Answer B: 887.5

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?

Answer B: ∫{0 to 1}(2x-(3x^2 - x^3))dx + ∫{1 to 2}((3x^2 - x^3) - 2x)dx

A particle moves along the x-axis with velocity given by v(t)=(t−1)e^(1-t) for time t≥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?

Answer B: ∫{0 to 2}(t-1)e^(1-t)dt

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

Answer C: 0.249

A traffic engineer developed the continuous function R, graphed above, to model the rate at which vehicles pass a certain intersection over an 88-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?

Answer C: 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]?

Answer C: 15/7

The base of a solid is the triangular region in the first quadrant bounded by the graph of y=5− (5/3)x 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?

Answer C: 25

The rate at which sand is poured into a bag is modeled by the function r given by r(t)=15t−2t^2, 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?

Answer C: 74/3

Let R be the region between the graph of y=arctanx, the x-axis, and the line x=1.5. Which of the following gives the area of region R?

Answer C: ∫{0 to arctan(1.5)}(1.5-tany)dy

The regions bounded by the graphs of y=−(2/π)x + 1 and y=cosx are shaded in the figure above. Which of the following gives the sum of the areas of the shaded regions?

Answer C: ∫{0 to π/2}(cosx-(-(2/π)x + 1))dx + ∫{π/2 to π}((-(2/π)x + 1) - cosx)dx

For time t≥0, the acceleration of an object moving in a straight line is given by a(t)=sin((t^2)/3). What is the net change in velocity from time t=0.75 to time t=2.25?

Answer D: 0.984

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?

Answer D: 3.012

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

Answer D: 87

The velocity of a particle moving along the x-axis is given by v(t)=−sin(t−π/4). Which of the following gives the total distance traveled by the particle over the time interval 0 ≤ t ≤ π?

Answer D: ∫{0 to π/4} v(t) dt - ∫{π/4 to π}v(t) dt