Exam 5: Key Concept- the Definite Integral

A car is moving along a straight road from A to B, starting from A at time t = 0.Below are graphs of the velocity and the acceleration plotted against time (positive direction is from A to B) A)The one on the right B)The one on the left. Which graph shows the velocity?

(Short Answer)

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Question 41

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=4+4 t^{2}$ for $0 \leq t \leq 6$ .Use $\Delta$ t = 1 to estimate distance during this time.(Average right- and left-hand sums).

(Short Answer)

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Question 42

Evaluate the definite integral $\int_{0}^{5}(5+2 x) d x$ .

(Short Answer)

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Question 43

If $\int_{a}^{b} f(x)=0$ , then f must have at least one zero between a and b (assume a $\neq$ b).

(Short Answer)

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Question 44

Evaluate $\int_{1}^{5} 3 x^{2} d x$

(Short Answer)

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Question 45

Let $\int_{0}^{8} f(x) d x=C$ .If f(x)is even, what is $\int_{-8}^{8} f(x) d x$ ?

(Multiple Choice)

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Question 46

The graph shown below is that of the velocity of an object (in meters/second).Find a lower estimate of the total distance traveled from t = 0 to t =5 seconds .

(Short Answer)

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Question 47

Estimate the area of the region under the curve $y=-x^{3}+2$ and above the x-axis for $0 \leq x \leq \sqrt[3]{2}$ .Round to 3 decimal places.

(Short Answer)

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Question 48

Let $\int_{0}^{10} f(x) d x=C$ .What is the average value of f(x)on the interval x = 0 to x = 10?

(Multiple Choice)

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Question 49

Rashmi and Tia both go running from 7:00am to 8:00am.Both women increase their velocity throughout the hour, both beginning at a rate of 9 mi/hr.at 7:00am and running at a rate of 13 mi/hr by 8:00am.Tia's velocity increases at an increasing rate and Rashmi's velocity increases at a decreasing rate.Who has the greatest average velocity? If they had the same average velocity, enter "same".

(Short Answer)

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Question 50

If a function is concave up, then the left-hand Riemann sums are always less than the right-hand Riemann sums with the same subdivisions, over the same interval.

(True/False)

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Question 51

Find the average value of $h(x)=3 x+3$ over [1, 3].

(Short Answer)

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Question 52

Estimate the area of the region between y = cos x, y = 3x, x = - $\pi$ /2, and x = 0.Round to 3 decimal places.

(Short Answer)

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Question 53

The average value of a function g on 0 F $\le$ x $\le$ 3 is a constant $\bar{g}$ given by $\bar{g}=\frac{1}{3-0} \int_{0}^{3} g(x) d x$ .Which of the following must be true?

(Multiple Choice)

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Question 54

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=2+5 t^{2}$ for $0 \leq t \leq 6$ .Use $\Delta$ t = 2 to estimate distance during this time.(Average right- and left-hand sums).

(Short Answer)

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Question 55

What is the value of $\int_{a}^{c}|f(x)| d x$ if the area of A = 9 and the area of B = -2? $What is the value of \int_{a}^{c}|f(x)| d x if the area of A = 9 and the area of B = -2?$

(Multiple Choice)

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Question 56

Estimate the area of the region above the curve $y=7 \cos x$ and below y = 7 for 0 $\le$ x $\le$ $\pi$ /2.Round to 2 decimal places.

(Short Answer)

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Question 57

A shop is open from 9am-7pm.The function r(t), graphed below, gives the rate at which customers arrive (in people/hour)at time t.Suppose that the salespeople can serve customers at a rate of 60 people per hour.When do people have to start waiting in line before being served? Answer to the nearest half-hour.You do not need to include "am" or "pm".

(Short Answer)

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Question 58

If $E(t)$ represents the energy consumed in a household in watts/month, what are the units of $\int_{a}^{3} E(t) d t$ ?

(Multiple Choice)

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Question 59

Find the average value of $f(x)=e^{3 x}$ over [0, 2].Round to the nearest whole number.

(Short Answer)

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Question 60

A car is moving along a straight road from A to B, starting from A at time t = 0.Below are graphs of the velocity and the acceleration plotted against time (positive direction is from A to B) A)The one on the right B)The one on the left. Which graph shows the velocity?

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=4+4 t^{2}$ for $0 \leq t \leq 6$ .Use $\Delta$ t = 1 to estimate distance during this time.(Average right- and left-hand sums).

Evaluate the definite integral $\int_{0}^{5}(5+2 x) d x$ .

If $\int_{a}^{b} f(x)=0$ , then f must have at least one zero between a and b (assume a $\neq$ b).

Evaluate $\int_{1}^{5} 3 x^{2} d x$

Let $\int_{0}^{8} f(x) d x=C$ .If f(x)is even, what is $\int_{-8}^{8} f(x) d x$ ?

The graph shown below is that of the velocity of an object (in meters/second).Find a lower estimate of the total distance traveled from t = 0 to t =5 seconds .

Estimate the area of the region under the curve $y=-x^{3}+2$ and above the x-axis for $0 \leq x \leq \sqrt[3]{2}$ .Round to 3 decimal places.

Let $\int_{0}^{10} f(x) d x=C$ .What is the average value of f(x)on the interval x = 0 to x = 10?

If a function is concave up, then the left-hand Riemann sums are always less than the right-hand Riemann sums with the same subdivisions, over the same interval.

Find the average value of $h(x)=3 x+3$ over [1, 3].

Estimate the area of the region between y = cos x, y = 3x, x = - $\pi$ /2, and x = 0.Round to 3 decimal places.

The average value of a function g on 0 F $\le$ x $\le$ 3 is a constant $\bar{g}$ given by $\bar{g}=\frac{1}{3-0} \int_{0}^{3} g(x) d x$ .Which of the following must be true?

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=2+5 t^{2}$ for $0 \leq t \leq 6$ .Use $\Delta$ t = 2 to estimate distance during this time.(Average right- and left-hand sums).

What is the value of $\int_{a}^{c}|f(x)| d x$ if the area of A = 9 and the area of B = -2? $What is the value of \int_{a}^{c}|f(x)| d x if the area of A = 9 and the area of B = -2?$

Estimate the area of the region above the curve $y=7 \cos x$ and below y = 7 for 0 $\le$ x $\le$ $\pi$ /2.Round to 2 decimal places.

If $E(t)$ represents the energy consumed in a household in watts/month, what are the units of $\int_{a}^{3} E(t) d t$ ?

Find the average value of $f(x)=e^{3 x}$ over [0, 2].Round to the nearest whole number.

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 5: Key Concept- the Definite Integral

A car is moving along a straight road from A to B, starting from A at time t = 0.Below are graphs of the velocity and the acceleration plotted against time (positive direction is from A to B) A)The one on the right B)The one on the left. Which graph shows the velocity?

At time t, in seconds, your velocity v, in meters/sec, is given by v(t)=4+4t2v(t)=4+4 t^{2}v(t)=4+4t2 for 0≤t≤60 \leq t \leq 60≤t≤6 .Use Δ\DeltaΔ t = 1 to estimate distance during this time.(Average right- and left-hand sums).

Evaluate the definite integral ∫05(5+2x)dx\int_{0}^{5}(5+2 x) d x∫05​(5+2x)dx .

If ∫abf(x)=0\int_{a}^{b} f(x)=0∫ab​f(x)=0 , then f must have at least one zero between a and b (assume a ≠\neq= b).

Evaluate ∫153x2dx\int_{1}^{5} 3 x^{2} d x∫15​3x2dx

Let ∫08f(x)dx=C\int_{0}^{8} f(x) d x=C∫08​f(x)dx=C .If f(x)is even, what is ∫−88f(x)dx\int_{-8}^{8} f(x) d x∫−88​f(x)dx ?

The graph shown below is that of the velocity of an object (in meters/second).Find a lower estimate of the total distance traveled from t = 0 to t =5 seconds .

Estimate the area of the region under the curve y=−x3+2y=-x^{3}+2y=−x3+2 and above the x-axis for 0≤x≤230 \leq x \leq \sqrt[3]{2}0≤x≤32​ .Round to 3 decimal places.

Let ∫010f(x)dx=C\int_{0}^{10} f(x) d x=C∫010​f(x)dx=C .What is the average value of f(x)on the interval x = 0 to x = 10?

If a function is concave up, then the left-hand Riemann sums are always less than the right-hand Riemann sums with the same subdivisions, over the same interval.

Find the average value of h(x)=3x+3h(x)=3 x+3h(x)=3x+3 over [1, 3].

Estimate the area of the region between y = cos x, y = 3x, x = - π\piπ /2, and x = 0.Round to 3 decimal places.

The average value of a function g on 0 F ≤\le≤ x ≤\le≤ 3 is a constant gˉ\bar{g}gˉ​ given by gˉ=13−0∫03g(x)dx\bar{g}=\frac{1}{3-0} \int_{0}^{3} g(x) d xgˉ​=3−01​∫03​g(x)dx .Which of the following must be true?

At time t, in seconds, your velocity v, in meters/sec, is given by v(t)=2+5t2v(t)=2+5 t^{2}v(t)=2+5t2 for 0≤t≤60 \leq t \leq 60≤t≤6 .Use Δ\DeltaΔ t = 2 to estimate distance during this time.(Average right- and left-hand sums).

What is the value of ∫ac∣f(x)∣dx\int_{a}^{c}|f(x)| d x∫ac​∣f(x)∣dx if the area of A = 9 and the area of B = -2?

Estimate the area of the region above the curve y=7cos⁡xy=7 \cos xy=7cosx and below y = 7 for 0 ≤\le≤ x ≤\le≤ π\piπ /2.Round to 2 decimal places.

If E(t)E(t)E(t) represents the energy consumed in a household in watts/month, what are the units of ∫a3E(t)dt\int_{a}^{3} E(t) d t∫a3​E(t)dt ?

Find the average value of f(x)=e3xf(x)=e^{3 x}f(x)=e3x over [0, 2].Round to the nearest whole number.

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=4+4 t^{2}$ for $0 \leq t \leq 6$ .Use $\Delta$ t = 1 to estimate distance during this time.(Average right- and left-hand sums).

Evaluate the definite integral $\int_{0}^{5}(5+2 x) d x$ .

If $\int_{a}^{b} f(x)=0$ , then f must have at least one zero between a and b (assume a $\neq$ b).

Evaluate $\int_{1}^{5} 3 x^{2} d x$

Let $\int_{0}^{8} f(x) d x=C$ .If f(x)is even, what is $\int_{-8}^{8} f(x) d x$ ?

Estimate the area of the region under the curve $y=-x^{3}+2$ and above the x-axis for $0 \leq x \leq \sqrt[3]{2}$ .Round to 3 decimal places.

Let $\int_{0}^{10} f(x) d x=C$ .What is the average value of f(x)on the interval x = 0 to x = 10?

Find the average value of $h(x)=3 x+3$ over [1, 3].

Estimate the area of the region between y = cos x, y = 3x, x = - $\pi$ /2, and x = 0.Round to 3 decimal places.

The average value of a function g on 0 F $\le$ x $\le$ 3 is a constant $\bar{g}$ given by $\bar{g}=\frac{1}{3-0} \int_{0}^{3} g(x) d x$ .Which of the following must be true?

At time t, in seconds, your velocity v, in meters/sec, is given by $v(t)=2+5 t^{2}$ for $0 \leq t \leq 6$ .Use $\Delta$ t = 2 to estimate distance during this time.(Average right- and left-hand sums).

What is the value of $\int_{a}^{c}|f(x)| d x$ if the area of A = 9 and the area of B = -2? $What is the value of \int_{a}^{c}|f(x)| d x if the area of A = 9 and the area of B = -2?$

Estimate the area of the region above the curve $y=7 \cos x$ and below y = 7 for 0 $\le$ x $\le$ $\pi$ /2.Round to 2 decimal places.

If $E(t)$ represents the energy consumed in a household in watts/month, what are the units of $\int_{a}^{3} E(t) d t$ ?

Find the average value of $f(x)=e^{3 x}$ over [0, 2].Round to the nearest whole number.