In Exercises 710, estimate the area of the region above

In Exercises 1 and 2, list the elements of the sequence. \(a_{k}=\frac{1}{2}\left(\frac{1}{2} k\right)^{2} \text { for } k=1,2,3,4, \ldots, 9,10\)

In Exercises 1 and 2, list the elements of the sequence. \(a_{k}=\frac{1}{4}\left(2+\frac{1}{4} k\right)^{2} \text { for } k=1,2,3,4, \ldots, 9,10\)

In Exercises 3–6, find the sum. \(\sum_{k=1}^{10} \frac{1}{2}(k+1)\)

In Exercises 3–6, find the sum. \(\sum_{k=1}^{n}(k+1)\)

In Exercises 3–6, find the sum. \(\sum_{k=1}^{10} \frac{1}{2}(k+1)^{2}\)

In Exercises 3–6, find the sum. \(\sum_{k=1}^{n} \frac{1}{2} k^{2}\)

A truck travels at an average speed of 57 mph for 4 hours. How far does it travel?

A pump working at 5 gal/min pumps for 2 hours. How many gallons are pumped?

Water flows over a spillway at a steady rate of 200 cubic feet per second. How many cubic feet of water pass over the spillway in 6 hours?

A county has a population density of 560 people per square mile in an area of 35,000 square miles. What is the population of the county?

In Exercises 1–4, explain how to represent the problem situation as an area question and then solve the problem. A train travels at 65 mph for 3 hours. How far does it travel?

In Exercises 1–4, explain how to represent the problem situation as an area question and then solve the problem. A pump working at 15 gal/min pumps for one-half hour. How many gallons are pumped?

In Exercises 1–4, explain how to represent the problem situation as an area question and then solve the problem. Water flows over a spillway at a steady rate of 150 cubic feet per second. How many cubic feet of water pass over the spillway in one hour?

In Exercises 1–4, explain how to represent the problem situation as an area question and then solve the problem. A city has a population density of 650 people per square mile in an area of 20 square miles. What is the population of the city?

An airplane travels at an average velocity of 640 kilometers per hour for 3 hours and 24 minutes. How far does the airplane travel?

A train travels at an average velocity of 24 miles per hour for 4 hours and 50 minutes. How far does the train travel?

In Exercises 7–10, estimate the area of the region above the x-axis and under the graph of the function from \(x=0 \text { to } x=5\).

In Exercises 7–10, estimate the area of the region above the x-axis and under the graph of the function from \(x=0 \text { to } x=5\).

In Exercises 7–10, estimate the area of the region above the x-axis and under the graph of the function from \(x=0 \text { to } x=5\).

In Exercises 7–10, estimate the area of the region above the x-axis and under the graph of the function from \(x=0 \text { to } x=5\).

In Exercises 11 and 12, use the 8 rectangles shown to approximate the area of the region below the graph of \(f(x)=10-x^{2}\) over the interval \([-1,3]\).

In Exercises 11 and 12, use the 8 rectangles shown to approximate the area of the region below the graph of \(f(x)=10-x^{2}\) over the interval \([-1,3]\).

In Exercises 13–16, partition the given interval into the indicated number of subintervals. \([0,2] ; \quad 4\)

In Exercises 13–16, partition the given interval into the indicated number of subintervals. \([0,2] ; \quad 8\)

In Exercises 13–16, partition the given interval into the indicated number of subintervals. \([1,4] ; \quad 6\)

In Exercises 13–16, partition the given interval into the indicated number of subintervals. \([1,5] ; \quad 8\)

In Exercises 17–20, complete the following. (a) Draw the graph of the function for x in the specified interval. Verify that the function is nonnegative in that interval. (b) On the graph in part (a), draw and shade the approximating rectangles for the RRAM using the specified partition. Compute the RRAM area estimate without using a calculator. (c) Repeat part (b) using the LRAM. (d) Average the RRAM and LRAM approximations from parts (b) and (c) to find an average estimate of the area. \(f(x)=x^{2} ;[0,4]\); 4 subintervals

In Exercises 17–20, complete the following. (a) Draw the graph of the function for x in the specified interval. Verify that the function is nonnegative in that interval. (b) On the graph in part (a), draw and shade the approximating rectangles for the RRAM using the specified partition. Compute the RRAM area estimate without using a calculator. (c) Repeat part (b) using the LRAM. (d) Average the RRAM and LRAM approximations from parts (b) and (c) to find an average estimate of the area. \(f(x)=x^{2}+2 ;[0,6]\); 6 subintervals

In Exercises 17–20, complete the following. (a) Draw the graph of the function for x in the specified interval. Verify that the function is nonnegative in that interval. (b) On the graph in part (a), draw and shade the approximating rectangles for the RRAM using the specified partition. Compute the RRAM area estimate without using a calculator. (c) Repeat part (b) using the LRAM. (d) Average the RRAM and LRAM approximations from parts (b) and (c) to find an average estimate of the area. \(f(x)=4 x-x^{2} ; \quad[0,4]\); 4 subintervals

In Exercises 17–20, complete the following. (a) Draw the graph of the function for x in the specified interval. Verify that the function is nonnegative in that interval. (b) On the graph in part (a), draw and shade the approximating rectangles for the RRAM using the specified partition. Compute the RRAM area estimate without using a calculator. (c) Repeat part (b) using the LRAM. (d) Average the RRAM and LRAM approximations from parts (b) and (c) to find an average estimate of the area. \(f(x)=x^{3} ;[0,3]\); 3 subintervals

In Exercises 21–28, find the definite integral by computing an area. (It may help to look at a graph of the function.) \(\int_{3}^{7} 5 d x\)

In Exercises 21–28, find the definite integral by computing an area. (It may help to look at a graph of the function.) \(\int_{-1}^{4} 6 d x\)

In Exercises 21–28, find the definite integral by computing an area. (It may help to look at a graph of the function.) \(\int_{0}^{5} 3 x d x\)

In Exercises 21–28, find the definite integral by computing an area. (It may help to look at a graph of the function.) \(\int_{1}^{7} 0.5 x d x\)

In Exercises 21–28, find the definite integral by computing an area. (It may help to look at a graph of the function.) \(\int_{1}^{4}(x+3) d x\)

In Exercises 21–28, find the definite integral by computing an area. (It may help to look at a graph of the function.) \(\int_{1}^{4}(3 x-2) d x\)

In Exercises 21–28, find the definite integral by computing an area. (It may help to look at a graph of the function.) \(\int_{-2}^{2} \sqrt{4-x^{2}} d x\)

In Exercises 21–28, find the definite integral by computing an area. (It may help to look at a graph of the function.) \(\int_{0}^{6} \sqrt{36-x^{2}} d x\)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{0}^{\pi} \sin x d x\)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{0}^{\pi}(\sin x+2) d x\)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{2}^{\pi+2} \sin (x-2) d x\)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{-\pi / 2}^{\pi / 2} \cos x d x\)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{0}^{\pi / 2} \sin x d x\)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{0}^{\pi / 2} \cos x d x\)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{0}^{\pi} 2 \sin x d x\) [Hint: All the rectangles are twice as tall.]

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{0}^{2 \pi} \sin \left(\frac{x}{2}\right) d x\) [Hint: All the rectangles are twice as wide.]

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{0}^{2 \pi}|\sin x| d x\)

It can be shown that the area enclosed between the x-axis and one arch of the sine curve is 2. Use this fact in Exercises 29–38 to compute the definite integral. (It may help to look at a graph of the function.) \(\int_{-\pi}^{3 \pi / 2}|\cos x| d x\)

In Exercises 39–42, find the integral, assuming that k is a number between 0 and 4. \(\int_{0}^{4}(k x+3) d x\)

In Exercises 39–42, find the integral, assuming that k is a number between 0 and 4. \(\int_{0}^{k}(4 x+3) d x\)

In Exercises 39–42, find the integral, assuming that k is a number between 0 and 4. \(\int_{0}^{4}(3 x+k) d x\)

In Exercises 39–42, find the integral, assuming that k is a number between 0 and 4. \(\int_{k}^{4}(4 x+3) d x\)

Writing to Learn Let \(g(x)=-f(x)\) where ƒ has nonnegative function values on an interval \([a, b]\). Explain why the area above the graph of g is the same as the area under the graph of ƒ in the same interval.

Writing to Learn Explain how you can find the area under the graph of \(f(x)=\sqrt{16-x^{2}} \text { from } x=0 \text { to } x=4\) from to by mental computation only.

Falling Ball Suppose a ball is dropped from a tower and its velocity after t seconds is always 32t feet per second. How far does the ball fall during the first 2 seconds?

Accelerating Automobile Suppose an automobile accelerates so that its velocity after t seconds is always 6t feet per second. How far does the car travel in the first 7 seconds?

Rock Toss A rock is thrown straight up from level ground. The velocity of the rock at any time t (sec) is \(v(t)=48-32 t \mathrm{ft} / \mathrm{sec}\). (a) Graph the velocity function. (b) At what time does the rock reach its maximum height? (c) Find how far the rock has traveled at its maximum height.

Rocket Launch A toy rocket is launched straight up from level ground. Its velocity function is \(f(t)=170-32 t\) feet per second, where t is the number of seconds after launch. (a) Graph the velocity function. (b) At what time does the rocket reach its maximum height? (c) Find how far the rocket has traveled at its maximum height.

Finding Distance Traveled as Area A ball is pushed off the roof of a three-story building. Table 10.3 gives the velocity (in feet per second) of the falling ball at 0.2-second intervals until it hits the ground 1.4 seconds later. (a) Draw a scatter plot of the data. (b) Find the approximate building height using RRAM areas as in Example 4. Use the fact that if the velocity function is always negative the distance traveled will be the same as if the absolute value of the velocity values were used.

Work Work is defined as force times distance. A full water barrel weighing 1250 pounds has a significant leak and must be lifted 35 feet. Table 10.4 displays the weight of the barrel measured after each 5 feet of movement. Find the approximate work in foot-pounds done in lifting the barrel 35 feet.

True or False When estimating the area under a curve using LRAM, the accuracy typically improves as the number n of subintervals is increased.

True or False The statement \(\lim _{x \rightarrow \infty} f(x)=L\) means that \(f(x)\) gets arbitrarily large as x gets arbitrarily close to L. It can be shown that the area of the region enclosed by the curve \(y=\sqrt{x}\), the x-axis, and the line \(x=9\) is 18. Use this fact in Exercises 53–56 to choose the correct answer. Do not use a calculator.

Multiple Choice \(\int_{0}^{9} 2 \sqrt{x} d x\) \(\begin{array}{l}\text{(A) 36}\\\text{(B) 27}\\\text{(C) 18}\\\text{(D) 9}\\\text{(E) 6}\end{array}\)

Multiple Choice \(\int_{0}^{9}(\sqrt{x}+5) d x\) \(\begin{array}{l}\text{(A) 14}\\\text{(B) 23}\\\text{(C) 33}\\\text{(D) 45}\\\text{(E) 63}\end{array}\)

Multiple Choice \(\int_{5}^{14}(\sqrt{x-5}) d x\) \(\begin{array}{l}\text{(A) 9}\\\text{(B) 13}\\\text{(C) 18}\\\text{(D) 23}\\\text{(E) 28}\end{array}\)

Multiple Choice \(\int_{0}^{3} \sqrt{3} x d x\) \(\begin{array}{l}\text{(A) 54}\\\text{(B) 18}\\\text{(C) 9}\\\text{(D) 6}\\\text{(E) 3}\end{array}\)

Group Activity You may have erroneously assumed that the function ƒ had to be positive in the definition of the definite integral. It is a fact that \(\int_{0}^{2 \pi} \sin x d x=0\). Use the definition of the definite integral to explain why this is so. What does this imply about \(\int_{0}^{1}(x-1) d x\)?

Area Under a Discontinuous Function Let \(f(x)=\left\{\begin{array}{ll} 1 & \text { if } x<2 \\ x & \text { if } x>2 \end{array}\right.\) (a) Draw a graph of ƒ. Determine its domain and range. (b) Writing to Learn How would you define the area under ƒ from \(x=0 \text { to } x=4\)? Does it make a difference if the function has no value at \(x=2\)?

Group Activity From what you know about definite integrals, decide whether each of the following statements is true or false for integrable functions (in general). Work with your classmates to justify your answers. \(\int_{a}^{b} f(x) d x+\int_{a}^{b} g(x) d x=\int_{a}^{b}(f(x)+g(x)) d x\)

Group Activity From what you know about definite integrals, decide whether each of the following statements is true or false for integrable functions (in general). Work with your classmates to justify your answers. \(\int_{a}^{b} 8 \cdot f(x) d x=8 \cdot \int_{a}^{b} f(x) d x\)

Group Activity From what you know about definite integrals, decide whether each of the following statements is true or false for integrable functions (in general). Work with your classmates to justify your answers. \(\int_{a}^{b} f(x) \cdot g(x) d x=\int_{a}^{b} f(x) d x \cdot \int_{a}^{b} g(x) d x\)

Group Activity From what you know about definite integrals, decide whether each of the following statements is true or false for integrable functions (in general). Work with your classmates to justify your answers. \(\int_{a}^{c} f(x) d x+\int_{c}^{b} f(x) d x=\int_{a}^{b} f(x) d x \text { for } a<c<b\)

Group Activity From what you know about definite integrals, decide whether each of the following statements is true or false for integrable functions (in general). Work with your classmates to justify your answers. \(\int_{a}^{b} f(x)=\int_{b}^{a} f(x)\)

Group Activity From what you know about definite integrals, decide whether each of the following statements is true or false for integrable functions (in general). Work with your classmates to justify your answers. \(\int_{a}^{a} f(x) d x=0\)