BC MCQ Set 30

Question 1 of 10 | Applications of Integration · Level 2

[Calculator] The average value of the function $f(x) = x^2 \sin x$ on the interval $[2, 4]$

$-0.686$

$0.686$

$-1.373$

$1.373$

$-2.746$

Question 2 of 10 | Integration and Accumulation of Change · Level 3

[Calculator] Let $g$ be the function given by $g(x) = \displaystyle\int_{0}^{x} \sin(t^2) d t$ for $-1 \leq x \leq 3$. On which of the following intervals is $g$ decreasing?

$-1 \leq x \leq 0$

$0 \leq x \leq 1.772$

$1.253 \leq x \leq 2.171$

$1.772 \leq x \leq 2.507$

$-1 \leq x \leq 3$

Question 3 of 10 | Applications of Integration · Level 3

If the region enclosed by the y-axis, the curve $y = 4 \sqrt{x}$, and the line $y = 8$ is revolved about the x-axis, the volume of the solid generated is

$\dfrac{32 \pi}{3}$

$128 \pi$

$\dfrac{128}{3}$

$128$

$\dfrac{128 \pi}{3}$

Question 4 of 10 | Applications of Integration · Level 3

[Calculator] Find the length of the curve $y = x^{\dfrac{3}{2}}$ from $x = 1$ to $x = 2$

$0$

$1.456$

$2.086$

$3.498$

$10.862$

Question 5 of 10 | Applications of Integration · Level 2

What is the average value of $y = \sin 2 x$ over the interval $[\dfrac{\pi}{4}, \dfrac{\pi}{3}]$?

$-\dfrac{6}{\pi}$

$-\dfrac{1}{6 \pi}$

$\dfrac{3}{\pi}$

$3 \pi$

$\dfrac{6}{\pi}$

Question 6 of 10 | Applications of Integration · Level 2

Which of the following integrals correctly corresponds to the area of the shaded region between $f(x) = 1 + x^2$ and $g(x) = 5$, in the first quadrant from $x = 1$ to $x = 2$?

$\displaystyle\int_{1}^{2} (x^2 - 4) d x$

$\displaystyle\int_{1}^{2} (4 - x^2) d x$

$\displaystyle\int_{1}^{5} (x^2 - 4) d x$

$\displaystyle\int_{1}^{5} (x^2 + 4) d x$

$\displaystyle\int_{1}^{5} (4 - x^2) d x$

Question 7 of 10 | Contextual Applications of Differentiation · Level 3

A particle's position is given by $s(t) = \sin t + 2 \cos t + \dfrac{t}{\pi} + 2$. The average velocity of the particle over $[0, 2 \pi]$

$-\dfrac{\pi + 1}{\pi}$

$-\dfrac{1}{3}$

$0$

$\dfrac{1}{\pi}$

$\dfrac{\pi + 1}{\pi}$

Question 8 of 10 | Applications of Integration · Level 3

A solid is generated when the region in the first quadrant enclosed by the graph of $y = (x^2 + 1)^3$, the line $x = 1$, the x-axis, and the y-axis is revolved about the x-axis. Its volume is found by evaluating which of the following integrals?

$\pi \displaystyle\int_{1}^{8} (x^2 + 1)^3 d x$

$\pi \displaystyle\int_{1}^{8} (x^2 + 1)^6 d x$

$\pi \displaystyle\int_{0}^{1} (x^2 + 1)^3 d x$

$\pi \displaystyle\int_{0}^{1} (x^2 + 1)^6 d x$

$2 \pi \displaystyle\int_{0}^{1} (x^2 + 1) d x$

Question 9 of 10 | Integration and Accumulation of Change · Level 1

$\displaystyle\int_{0}^{\dfrac{\pi}{2}} \cos x d x =$

$-\pi$

$-1$

$0$

$1$

$\pi$

Question 10 of 10 | Integration and Accumulation of Change · Level 1

$\dfrac{d}{d x} \displaystyle\int_{0}^{x} \sin(t) d t =$

$\sin t$

$\cos t$

$-\cos x$

$\sin x$

$\cos x$

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Reference Sheet

Area & Circumference

Circle$A = \pi r^2$, $C = 2\pi r$

Rectangle$A = lw$

Triangle$A = \tfrac{1}{2}bh$

Trapezoid$A = \tfrac{1}{2}(b_1+b_2)h$

Volume

Box$V = lwh$

Cylinder$V = \pi r^2 h$

Sphere$V = \tfrac{4}{3}\pi r^3$

Cone$V = \tfrac{1}{3}\pi r^2 h$

Pyramid$V = \tfrac{1}{3}lwh$

Triangles

Pythagorean Thm$a^2 + b^2 = c^2$

30-60-90sides: $1,\, \sqrt{3},\, 2$

45-45-90sides: $1,\, 1,\, \sqrt{2}$

Triangle Anglessum $= 180°$

Other Facts

Circle Degrees$360° = 2\pi \text{ rad}$

Exterior Angle= sum of non-adjacent interior angles

The number of degrees of arc in a circle is 360. The number of radians of arc in a circle is $2\pi$.

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