정적분의 뜻과 성질

[T] Using your velocity equation from the previous exercise, find the corresponding distance equation, assuming your initial distance is 0 mi. How far did you travel while you accelerated your car? (Hint: You will need to convert time units.)

[T] The number of hamburgers sold at a restaurant throughout the day is given in the following table, with the accompanying graph plotting the best cubic fit to the data, $b(t)=0.12{t}^{3}-2.13{t}^{2}+12.13t+3.91,$ with $t=0$ corresponding to 9 a.m. and $t=12$ corresponding to 9 p.m. Compute the average value of $b(t)$ to estimate the average number of hamburgers sold per hour.

[T] An athlete runs by a motion detector, which records her speed, as displayed in the following table. The best linear fit to this data, $\ell (t)=-0.068t+5.14,$ is shown in the accompanying graph. Use the average value of $\ell (t)$ between $t=0$ and $t=40$ to estimate the runner’s average speed.

What is “total signed area”?

What is “displacement”?

What is $\displaystyle \int_3^3 \sin x\ dx$?

Give a single definite integral that has the same value as

$\displaystyle \int_0^1(2x+3)\ dx + \int_1^2 (2x+3)\ dx$.

In Exercises 5–10, a graph of a function $f(x)$ is given. Using the geometry of the graph, evaluate the definite integrals.

(a) $\displaystyle \int_0^1 (-2x+4)\ dx$ (b) $\displaystyle \int_0^2 (-2x+4)\ dx$ (c) $\displaystyle \int_0^3 (-2x+4)\ dx$

(a) $\displaystyle \int_1^3 (-2x+4)\ dx$ (b) $\displaystyle \int_2^4 (-2x+4)\ dx$ (c) $\displaystyle \int_0^1 (-6x+12)\ dx$

In Exercises 5–10, a graph of a function $f(x)$ is given. Using the geometry of the graph, evaluate the definite integrals.

(a) $\displaystyle \int_0^2 f(x)\ dx$ (b) $\displaystyle \int_0^3 f(x)\ dx$ (c) $\displaystyle \int_0^5 f(x)\ dx$

(a) $\displaystyle \int_2^5 f(x)\ dx$ (b) $\displaystyle \int_5^3 f(x)\ dx$ (c) $\displaystyle \int_0^3 -2f(x)\ dx$

In Exercises 5–10, a graph of a function $f(x)$ is given. Using the geometry of the graph, evaluate the definite integrals.

(a) $\displaystyle \int_0^2 f(x)\ dx$ (b) $\displaystyle \int_2^4 f(x)\ dx$ (c) $\displaystyle \int_2^4 2f(x)\ dx$

(a) $\displaystyle \int_0^1 4x\ dx$ (b) $\displaystyle \int_2^3 (2x-4)\ dx$ (c) $\displaystyle \int_2^3 (4x-8)\ dx$

In Exercises 5–10, a graph of a function $f(x)$ is given. Using the geometry of the graph, evaluate the definite integrals.

(a) $\displaystyle \int_0^1 (x-1)\ dx$ (b) $\displaystyle \int_0^2 (x-1)\ dx$ (c) $\displaystyle \int_0^3 (x-1)\ dx$

(a) $\displaystyle \int_2^3 (x-1)\ dx$ (b) $\displaystyle \int_1^4 (x-1)\ dx$ (c) $\displaystyle \int_1^4 \big((x-1)+1\big)\ dx$

In Exercises 5–10, a graph of a function $f(x)$ is given. Using the geometry of the graph, evaluate the definite integrals.

(a) $\displaystyle \int_0^2 f(x)\ dx$ (b) $\displaystyle \int_2^4 f(x)\ dx$

(a) $\displaystyle \int_0^4 f(x)\ dx$ (b) $\displaystyle \int_0^4 5f(x)\ dx$

In Exercises 5–10, a graph of a function $f(x)$ is given. Using the geometry of the graph, evaluate the definite integrals.

(a) $\displaystyle \int_0^5 f(x)\ dx$ (b) $\displaystyle \int_3^7 f(x)\ dx$

(a) $\displaystyle \int_0^0 f(x)\ dx$ (b) $\displaystyle \int_a^b f(x)\ dx$, where

$0\leq a\leq b\leq 10$

In Exercises 11–14, a graph of a function $f(x)$ is given; the numbers inside the shaded regions give the area of that region. Evaluate the definite integrals using this area information.

(a) $\displaystyle \int_0^1 f(x)\ dx$ (b) $\displaystyle \int_0^2 f(x)\ dx$

(a) $\displaystyle \int_0^3 f(x)\ dx$ (b) $\displaystyle \int_1^2 -3f(x)\ dx$

In Exercises 11–14, a graph of a function $f(x)$ is given; the numbers inside the shaded regions give the area of that region. Evaluate the definite integrals using this area information.

(a) $\displaystyle \int_0^2 f(x)\ dx$ (b) $\displaystyle \int_2^4 f(x)\ dx$

(a) $\displaystyle \int_0^4 f(x)\ dx$ (b) $\displaystyle \int_0^1 f(x)\ dx$

In Exercises 11–14, a graph of a function $f(x)$ is given; the numbers inside the shaded regions give the area of that region. Evaluate the definite integrals using this area information.

(a) $\displaystyle \int_{-2}^{-1} f(x)\ dx$ (b) $\displaystyle \int_1^2 f(x)\ dx$

(a) $\displaystyle \int_{-1}^1 f(x)\ dx$ (b) $\displaystyle \int_0^1 f(x)\ dx$

In Exercises 11–14, a graph of a function $f(x)$ is given; the numbers inside the shaded regions give the area of that region. Evaluate the definite integrals using this area information.

(a) $\displaystyle \int_{0}^{2} 5x^2\ dx$ (b) $\displaystyle \int_0^2 (x^2+3)\ dx$

(a) $\displaystyle \int_{1}^3 (x-1)^2\ dx$ (b) $\displaystyle \int_2^4 \big((x-2)^2+5\big)\ dx$

In Exercises 15–16, a graph of the velocity function of an object moving in a straight line is given. Answer the questions based on that graph.

(a) What is the object's maximum velocity? (b) What is the object's maximum displacement? (c) What is the object's total displacement on $[0,3]$?

In Exercises 15–16, a graph of the velocity function of an object moving in a straight line is given. Answer the questions based on that graph.

(a) What is the object's maximum velocity? (b) What is the object's maximum displacement? (c) What is the object's total displacement on $[0,5]$?

An object is thrown straight up with a velocity, in ft/s, given by $v(t) = -32t+64$, where $t$ is in seconds, from a height of 48 feet.

(a) What is the object's maximum velocity? (b) What is the object's maximum displacement? (c) When does the maximum displacement occur? (d) When will the object reach a height of 0? (Hint: find when the displacement is $-48$ft.)