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A particle has velocity $v(t) = 2t + 7$ and starts at $s(0) = 4$ . Find its position at $t = 5$ .
A particle's velocity is $v(t) = t - 4$ m/s. Find the total distance it travels from $t = 0$ to $t = 6$ , in meters.
$v(t)$ is a velocity in m/s. What does $\displaystyle\int_0^{ 9 } |v(t)|\,dt$ give?
1. The total distance traveled from $t = 0$ to $t = 9$
2. The displacement from $t = 0$ to $t = 9$
3. The average speed on $[0, 9]$
$v(t)$ is a velocity in m/s. What does $\displaystyle\int_0^{ 2 } |v(t)|\,dt$ give?
1. The total distance traveled from $t = 0$ to $t = 2$
2. The displacement from $t = 0$ to $t = 2$
3. The average speed on $[0, 2]$
$v(t)$ is a velocity in m/s. What does $\displaystyle\int_0^{ 6 } |v(t)|\,dt$ give?
1. The total distance traveled from $t = 0$ to $t = 6$
2. The displacement from $t = 0$ to $t = 6$
3. The average speed on $[0, 6]$
$v(t)$ is a velocity in m/s. What does $\displaystyle\int_0^{ 8 } |v(t)|\,dt$ give?
1. The total distance traveled from $t = 0$ to $t = 8$
2. The displacement from $t = 0$ to $t = 8$
3. The average speed on $[0, 8]$
$v(t)$ is a velocity in m/s. What does $\displaystyle\int_0^{ 7 } |v(t)|\,dt$ give?
1. The total distance traveled from $t = 0$ to $t = 7$
2. The displacement from $t = 0$ to $t = 7$
3. The average speed on $[0, 7]$
$v(t)$ is a velocity in m/s. What does $\displaystyle\int_0^{ 5 } |v(t)|\,dt$ give?
1. The total distance traveled from $t = 0$ to $t = 5$
2. The displacement from $t = 0$ to $t = 5$
3. The average speed on $[0, 5]$
$v(t)$ is a velocity in m/s. What does $\displaystyle\int_0^{ 4 } |v(t)|\,dt$ give?
1. The total distance traveled from $t = 0$ to $t = 4$
2. The displacement from $t = 0$ to $t = 4$
3. The average speed on $[0, 4]$
$v(t)$ is a velocity in m/s. What does $\displaystyle\int_0^{ 3 } |v(t)|\,dt$ give?
1. The total distance traveled from $t = 0$ to $t = 3$
2. The displacement from $t = 0$ to $t = 3$
3. The average speed on $[0, 3]$
Find the area of the region between $y = 5x$ and $y = x^2$ .
Find the area of the region between $y = x^2$ and $y = 4x - x^2$ .
Find the area between $y = 1\cos x$ and $y = 1\sin x$ for $0 \le x \le \dfrac{\pi}{4}$ . Give an exact value.
Find the area of the region between $x = y^2$ and $x = 9$ , integrating with respect to $y$ .
Find the area of the region between $x = y^2$ and $x = 2y + 8$ .
Which integral gives the area between $x = y^2$ and $x = 16$ ?
1. $\displaystyle\int_{-4}^{ 4 } (16 - y^2)\,dy$
2. $\displaystyle\int_0^{ 16 } (16 - y^2)\,dy$
3. $\displaystyle\int_{-4}^{ 4 } (y^2 - 16)\,dy$
Which integral gives the area between $x = y^2$ and $x = 4$ ?
1. $\displaystyle\int_{-2}^{ 2 } (4 - y^2)\,dy$
2. $\displaystyle\int_0^{ 4 } (4 - y^2)\,dy$
3. $\displaystyle\int_{-2}^{ 2 } (y^2 - 4)\,dy$
Which integral gives the area between $x = y^2$ and $x = 9$ ?
1. $\displaystyle\int_{-3}^{ 3 } (9 - y^2)\,dy$
2. $\displaystyle\int_0^{ 9 } (9 - y^2)\,dy$
3. $\displaystyle\int_{-3}^{ 3 } (y^2 - 9)\,dy$
Which integral gives the area between $x = y^2$ and $x = 25$ ?
1. $\displaystyle\int_{-5}^{ 5 } (25 - y^2)\,dy$
2. $\displaystyle\int_0^{ 25 } (25 - y^2)\,dy$
3. $\displaystyle\int_{-5}^{ 5 } (y^2 - 25)\,dy$
A solid's base is the region under $y = \sqrt{x}$ for $0 \le x \le 3$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.
Same base (under $y = \sqrt{x}$ , $0 \le x \le 2$ ), but the cross-sections are semicircles whose diameter lies in the base. Find the volume.
A solid's base is the triangle under $y = 4x$ for $0 \le x \le 4$ . Cross-sections perpendicular to the $x$ -axis are squares. Find its volume.
The region under $y = \sqrt{x}$ , $0 \le x \le 1$ , is revolved about the $x$ -axis. Find the volume.
The region under $y = x$ , $0 \le x \le 4$ , is revolved about the $x$ -axis (a cone). Find the volume.
The region under $y = e^x$ , $0 \le x \le 1$ , is revolved about the $x$ -axis. Find the volume.
The region between $y = 3x$ and $y = x^2$ is revolved about the $x$ -axis. Find the volume.
The region under $y = \sqrt{x}$ , $0 \le x \le 9$ , is revolved about the line $y = -1$ . Find the volume.
The region between $y = 16$ and $y = x^2$ is revolved about the $x$ -axis. Which integral is its volume?
1. $\pi\displaystyle\int_{-4}^{ 4 } \big(256 - x^4\big)\,dx$
2. $\pi\displaystyle\int_{-4}^{ 4 } \big(16 - x^2\big)^2\,dx$
3. $\pi\displaystyle\int_{-4}^{ 4 } \big(16 - x^2\big)\,dx$
The region between $y = 4$ and $y = x^2$ is revolved about the $x$ -axis. Which integral is its volume?
1. $\pi\displaystyle\int_{-2}^{ 2 } \big(16 - x^4\big)\,dx$
2. $\pi\displaystyle\int_{-2}^{ 2 } \big(4 - x^2\big)^2\,dx$
3. $\pi\displaystyle\int_{-2}^{ 2 } \big(4 - x^2\big)\,dx$
The region between $y = 9$ and $y = x^2$ is revolved about the $x$ -axis. Which integral is its volume?
1. $\pi\displaystyle\int_{-3}^{ 3 } \big(81 - x^4\big)\,dx$
2. $\pi\displaystyle\int_{-3}^{ 3 } \big(9 - x^2\big)^2\,dx$
3. $\pi\displaystyle\int_{-3}^{ 3 } \big(9 - x^2\big)\,dx$

Answers

$64$
$10$
The total distance traveled from $t = 0$ to $t = 9$
$\frac{125}{6}$
$\frac{8}{3}$
$-1 + \sqrt{2}$
$36$
$36$
$\displaystyle\int_{-4}^{ 4 } (16 - y^2)\,dy$
$\frac{9}{2}$
$\frac{\pi}{4}$
$\frac{1024}{3}$
$\frac{\pi}{2}$
$\frac{64 \pi}{3}$
$\frac{\pi \left(-1 + e^{2}\right)}{2}$
$\frac{162 \pi}{5}$
$\frac{153 \pi}{2}$
$\pi\displaystyle\int_{-4}^{ 4 } \big(256 - x^4\big)\,dx$