적분 전체 문제

[T] The graph below plots the quadratic $p(t)=6.48{t}^{2}-80.3\,1t+585.69$ against the data in preceding table, normalized so that $t=0$ corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of p over $[0,37].$

[T] The graph below plots the cubic $p(t)=0.07{t}^{3}+2.42{t}^{2}-25.63t+521.23$ against the data in the preceding table, normalized so that $t=0$ corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of p over $[0,37].$

[T] Suppose you go on a road trip and record your speed at every half hour, as compiled in the following table. The best quadratic fit to the data is $q(t)=5{x}^{2}-11x+49,$ shown in the accompanying graph. Integrate q to estimate the total distance driven over the 3 hours.

[T] The accompanying graph plots the best quadratic fit, $a(t)=-0.70{t}^{2}+1.44t+10.44,$ to the data from the preceding table. Compute the average value of $a(t)$ to estimate the average acceleration between $t=0$ and $t=5.$

[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.

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 $[0,5]$?

In Exercises 51–54, sketch the given functions and find the area of the enclosed region.

$y=2x$, $y=5x$, and $x= 3$.

In Exercises 51–54, sketch the given functions and find the area of the enclosed region.

$y=-x+1$, $y=3x+6$, $x=2$ and $x= -1$.