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A car drives at 30 mph for 2 hours, then at 57 mph until hour 5. How far does it travel, in miles?
From rest, a cart's velocity is $v(t) = 4t$ m/s. How far does it go in the first 6 seconds, in meters?
A particle's velocity is $v(t) = t - 1$ m/s. Find its displacement from $t = 0$ to $t = 8$ , in meters (areas below the axis count as negative).
Estimate $\displaystyle\int_0^{ 5 } x^2\,dx$ with a left Riemann sum of 5 subintervals of width 1.
Estimate $\displaystyle\int_0^4 (x^2 + 4)\,dx$ with a midpoint sum of 2 subintervals.
$f$ is increasing on $[0, 7]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 7 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact
$f$ is increasing on $[0, 9]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 9 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact
$f$ is increasing on $[0, 3]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 3 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact
$f$ is increasing on $[0, 5]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 5 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact
$f$ is increasing on $[0, 2]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 2 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact
$f$ is increasing on $[0, 6]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 6 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact
$f$ is increasing on $[0, 4]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 4 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact
$f$ is increasing on $[0, 8]$ . How does a left Riemann sum compare with $\displaystyle\int_0^{ 8 } f(x)\,dx$ ?
1. It underestimates: each rectangle uses the lowest height on its piece
2. It overestimates
3. It is exact
Use geometry: find $\displaystyle\int_{-6}^{ 6 } \sqrt{ 36 - x^2}\,dx$ .
$\displaystyle\int_1^3 f(x)\,dx = 7$ and $\displaystyle\int_3^7 f(x)\,dx = -4$ . Find $\displaystyle\int_7^1 f(x)\,dx$ .
$\displaystyle\int_0^{ 4 } f(x)\,dx = 3$ . Find $\displaystyle\int_0^{ 4 } \big(3f(x) + 2\big)\,dx$ .
Find the antiderivative $F$ of $f(x) = 4x^{ 3 } + 9$ with $F(0) = 0$ .
Find the antiderivative $F$ of $f(x) = \cos x + e^x$ with $F(0) = 6$ .
Find the antiderivative $F$ of $f(x) = \dfrac{1}{x}$ , for $x > 0$ , with $F(1) = -9$ .
Evaluate $\displaystyle\int_0^{ 5 } x^2\,dx$ .
Evaluate $\displaystyle\int_0^{\pi/5} \cos x\,dx$ . Give an exact value.
Evaluate $\displaystyle\int_{ -1 }^{ 4 } (2x + 9)\,dx$ .
Find the average value of $f(x) = x^2$ on $[0, 3]$ .
$v(t) = 3t + 6$ m/s. Find the average velocity on $[0, 8]$ , in m/s.
Find the average value of $f(x) = 6\sin x$ on $[0, \pi]$ . Give an exact value.
$g(x) = \displaystyle\int_{ 4 }^{x} (t^{ 2 } + 2)\,dt$ . Find $g'(x)$ .
$g(x) = \displaystyle\int_0^{x^{ 4 } } \cos t\,dt$ . Find $g'(x)$ .
Here is the graph of $y = f(t)$ . Let $g(x) = \displaystyle\int_{-5}^{x} f(t)\,dt$ . Tap the point of this graph at the $x$ -value where $g$ has a local maximum.
$F' = f$ , $F(0) = -1$ and $\displaystyle\int_0^{ 3 } f(x)\,dx = 0$ . Find $F(3)$ .
A tank holds 37 L. Water flows in at $r(t) = 5 + 1t$ L/min. How much water is in the tank after 4 minutes, in liters?
$v(t)$ is a particle's velocity in m/s. What does $\displaystyle\int_0^{ 7 } v(t)\,dt$ give?
1. Its displacement (change in position) from $t = 0$ to $t = 7$ , in meters
2. The total distance it travels, in meters
3. Its velocity at $t = 7$
$v(t)$ is a particle's velocity in m/s. What does $\displaystyle\int_0^{ 5 } v(t)\,dt$ give?
1. Its displacement (change in position) from $t = 0$ to $t = 5$ , in meters
2. The total distance it travels, in meters
3. Its velocity at $t = 5$
$v(t)$ is a particle's velocity in m/s. What does $\displaystyle\int_0^{ 4 } v(t)\,dt$ give?
1. Its displacement (change in position) from $t = 0$ to $t = 4$ , in meters
2. The total distance it travels, in meters
3. Its velocity at $t = 4$
$v(t)$ is a particle's velocity in m/s. What does $\displaystyle\int_0^{ 3 } v(t)\,dt$ give?
1. Its displacement (change in position) from $t = 0$ to $t = 3$ , in meters
2. The total distance it travels, in meters
3. Its velocity at $t = 3$
$v(t)$ is a particle's velocity in m/s. What does $\displaystyle\int_0^{ 6 } v(t)\,dt$ give?
1. Its displacement (change in position) from $t = 0$ to $t = 6$ , in meters
2. The total distance it travels, in meters
3. Its velocity at $t = 6$
$v(t)$ is a particle's velocity in m/s. What does $\displaystyle\int_0^{ 2 } v(t)\,dt$ give?
1. Its displacement (change in position) from $t = 0$ to $t = 2$ , in meters
2. The total distance it travels, in meters
3. Its velocity at $t = 2$
$v(t)$ is a particle's velocity in m/s. What does $\displaystyle\int_0^{ 9 } v(t)\,dt$ give?
1. Its displacement (change in position) from $t = 0$ to $t = 9$ , in meters
2. The total distance it travels, in meters
3. Its velocity at $t = 9$
$v(t)$ is a particle's velocity in m/s. What does $\displaystyle\int_0^{ 8 } v(t)\,dt$ give?
1. Its displacement (change in position) from $t = 0$ to $t = 8$ , in meters
2. The total distance it travels, in meters
3. Its velocity at $t = 8$

Answers

$231$
$72$
$24$
$30$
$36$
It underestimates: each rectangle uses the lowest height on its piece
$18 \pi$
$-3$
$17$
$x^{4} + 9 x$
$e^{x} + \sin{\left(x \right)} + 5$
$\ln{\left(x \right)} - 9$
$\frac{125}{3}$
$\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}$
$60$
$3$
$18$
$\frac{12}{\pi}$
$x^{2} + 2$
$4 x^{3} \cos{\left(x^{4} \right)}$
$(3, 0)$
$-1$
$65$
Its displacement (change in position) from $t = 0$ to $t = 7$ , in meters