?Using properties of integrals Use the value of the first integral I to evaluate the two

Use graphs to evaluate \(\int _{0}^{2\pi}\ sin\ x\ dx\) and \(\int _{0}^{2\pi}\ cos\ x\ dx\).

When does the net area of a region equal the area of a region? When does the net area of a region differ from the area of a region?

Suppose that f(x) < 0 on the interval [a, b]. Using Riemann sums, explain why the definite integral \(\int _{a}^{b}\ f(x)\ dx\) is negative.

How do you interpret geometrically the definite integral of a function that changes sign on the interval of integration?

Explain what net area means.

Explain how the notation for Riemann sums. \(\sum _{k=1}^n\ f(\overline x_k) \triangle x\), corresponds to the notation for the definite integral, \(\int _{a}^{b}\ f(x)\ dx\).

Give a geometrical explanation of why \(\int _{a}^{b}\ f(x)\ dx=0\).

Use Table 5.3 to rewrite \(\int _{1}^{6}\ (2x^3-4x)\ dx\) as the sum of two integrals.

Use geometry to find a formula for \(\int _{0}^{a}\ x\ dx\), in terms of a.

If f is continuous on [a, b] and \(\int _{a}^{b}\ |f(x)|\ dx=0\), what can you conclude about f?

Approximating net area The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. f(x) = -2x - 1; [0, 4]

Approximating net area The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. \(f(x)=-4-x^3;\ [3,\ 7]\)

Approximating net area The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. \(f(x)=\sin 2x;\ [\pi/2,\ \pi]\)

Approximating net area The following functions are negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. \(f(x)=x^3-1;\ [-2,\ 0]\)

Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. f(x) = 4 - 2x; [0, 4]

Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. \(f(x)=8-2x^2;\ [0,\ 4]\)

Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. \(f(x)=\sin 2x;\ [0,\ 3\pi/4]\)

Approximating net area The following functions are positive and negative on the given interval. a. Sketch the function on the given interval. b. Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4. c. Use the sketch in part (a) to show which intervals of [a, b] make positive and negative contributions to the net area. \(f(x)=x^3;\ [-1,\ 2]\)

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums of a function f an [a, b]. Identify f and express the limits as a definite integral. \(\lim _{\triangle \rightarrow 0}\ \sum _{k=1}^{n}\ (\overline x_k^2+1)\ \triangle x_k;\ [0,\ 2]\)

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums of a function f an [a, b]. Identify f and express the limits as a definite integral. \(\lim _{\triangle \rightarrow 0}\ \sum _{k=1}^{n}\ (4-\overline x_k^2)\ \triangle x_k;\ [-2,\ 2]\)

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums of a function f an [a, b]. Identify f and express the limits as a definite integral. \(\lim _{\triangle \rightarrow 0}\ \sum _{k=1}^{n}\ \overline x_k\ ln\ \overline x_k\ \triangle x_k;\ [1,\ 2]\)

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums of a function f an [a, b]. Identify f and express the limits as a definite integral. \(\lim _{\triangle \rightarrow 0}\ \sum _{k=1}^{n}\ |\overline x_k^2-1|\ \triangle x_k;\ [-2,\ 2]\)

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. \(\int _0^4\ (8-2x)\ dx\)

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. \(\int _{-4}^2\ (2x+4)\ dx\)

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. \(\int _{-1}^2\ (-|x|)\ dx\)

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. \(\int _0 ^2\ (1-|x|)\ dx\)

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. \(\int _0^4\ \sqrt{16-x^2}\ dx\)

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. \(\int _{-1}^3\ \sqrt{4-(x-1)^2}\ dx\)

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. \(\int _0^4\ f(x)\ dx\) where \(f(x)=\left\{\begin{array}{ll} 5 & \text { if } x \leq 2 \\ 3 x-1 & \text { if } x>2 \end{array}\right.\)

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result. \(\int _1^{10}\ g(x)\ dx\) where \(g(x)=\left\{\begin{array}{ll} 4 x & \text { if } 0 \leq x \leq 2 \\ -8 x+16 & \text { if } 2<x \leq 3 \\ -8 & \text { if } x>3 \end{array}\right.\)

Net area from graphs The figure shows the areas of regions bounded by the graph f and the x-axis. Evaluate the following integrals. \(\int _0 ^a\ f(x)\ dx\)

Net area from graphs The figure shows the areas of regions bounded by the graph f and the x-axis. Evaluate the following integrals. \(\int _0^b\ f(x)\ dx\)

Net area from graphs The figure shows the areas of regions bounded by the graph f and the x-axis. Evaluate the following integrals. \(\int _a^c\ f(x)\ dx\)

Net area from graphs The figure shows the areas of regions bounded by the graph f and the x-axis. Evaluate the following integrals. \(\int _0^c\ f(x)\ dx\)

Net area from graphs The accompanying figure shows four regions bounded by the graph of \(y=x \sin x:\ R_1,\ R_2,\ R_3,\) and \(R_4\), whose areas are \(1,\ \pi-1,\ \pi+1\), and \(2\pi-1\), respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. \(\int _0^{\pi}\ x \sin x\ dx\)

Net area from graphs The accompanying figure shows four regions bounded by the graph of \(y=x \sin x:\ R_1,\ R_2,\ R_3,\) and \(R_4\), whose areas are \(1,\ \pi-1,\ \pi+1\), and \(2\pi-1\), respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. \(\int _0^{3\pi/2}\ x \sin x\ dx\)

Net area from graphs The accompanying figure shows four regions bounded by the graph of \(y=x \sin x:\ R_1,\ R_2,\ R_3,\) and \(R_4\), whose areas are \(1,\ \pi-1,\ \pi+1\), and \(2\pi-1\), respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. \(\int _0^{2\pi}\ x \sin x\ dx\)

Net area from graphs The accompanying figure shows four regions bounded by the graph of \(y=x \sin x:\ R_1,\ R_2,\ R_3,\) and \(R_4\), whose areas are \(1,\ \pi-1,\ \pi+1\), and \(2\pi-1\), respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. \(\int _{\pi/2}^{2\pi}\ x \sin x\ dx\)

Properties of integrals Use only the fact that \(\int _0^4\ 3x(4-x)\ dx=32\) and the definitions and properties of integrals to evaluate the following integrals, if possible. a. \(\int _4^0\ 3x(4-x)\ dx\) b. \(\int _0^4\ x(x-4)\ dx\) c. \(\int _4^0\ 6x(4-x)\ dx\) d. \(\int _0^8\ 3x(4-x)\ dx\)

Properties of integrals Suppose \(\int _1^4\ f(x)\ dx=8\) and \(\int _1^6\ f(x)\ dx=5\). Evaluate the following integrals. a. \(\int _1^4\ (-3\ f(x))\ dx\) b. \(\int _1^4\ 3f(x)\ dx\) c. \(\int _0^4\ 12f(x)\ dx\) d. \(\int _4^6\ 3f(x)\ dx\)

Properties of integrals Suppose \(\int _0^3\ f(x)\ dx=2,\ \int _3^6\ f(x)\ dx=-5\), and \(\int _3^6\ g(x)\ dx=1\). Evaluate the following integrals. a. \(\int _0^3\ 5f(x)\ dx\) b. \(\int _3^6\ [-3g(x)]\ dx\) c. \(\int _3^6\ (3f(x)-g(x))\ dx\) d. \(\int _6 ^3\ [f(x)+2g(x)]\ dx\)

Properties of integrals Suppose that \(f(x)\ \geq\ 0\) on [0, 2], \(f(x)\ \leq\ 0\) on [2, 5], \(\int _0^2\ f(x)\ dx=6\), and \(\int _2^5\ f(x)\ dx=-8\). Evaluate the following integrals. a. \(\int _0^5\ f(x)\ dx\) b. \(\int _0^5\ |f(x)|\ dx\) c. \(\int _2 ^5\ 4|f(x)|\ dx\) d. \(\int _0^5\ (f(x)+|f(x)|)\ dx\)

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. \(I=\int _0^1\ (x^3-2x)\ dx\ =\ -\frac{3}{4}\) a. \(\int _0^1\ (4x-2x^3)\ dx\) b. \(\int _1^0\ (2x-x^3)\ dx\)

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. \(I=\int _0^{\pi/2}\ (\cos \theta\-2 \sin \theta)\ d \theta=-1\) a. \(\int _0^{\pi/2}\ (2 \sin \theta- \cos \theta)\ d \theta\) b. \(\int _{\pi/2}^0\ (4 \cos \theta-8 \sin \theta)\ d \theta\)

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. \(\int _0^2\ (2x+1)\ dx\)

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. \(\int _1^5\ (1-x)\ dx\)

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. \(\int _3^7\ (4x+6)\ dx\)

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. \(\int _0^2\ (x^2-1)\ dx\)

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. \(\int _1^4\ (x^2-1)\ dx\)

Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1. \(\int _0^2\ 4x^3\ dx\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If f is a constant function on the interval [a, b], then the right and left Riemann sums give the exact value of \(\int _a^b\ f(x)\ dx\) for any n. b. If f is a linear function on the interval [a, b], then a midpoint Riemann sum gives the exact value of \(\int _a^b\ f(x)\ dx\) for any n. c. \(\int _0^{2\pi/a} \sin ax\ dx\ =\ \int _0^{2\pi/a} \cos ax\ dx\ =\ 0\) (Hint: Graph the functions and use properties of trigonometric functions). d. If \(\int _a^b\ f(x)\ dx\ =\ \int _b^a\ f(x)\ dx\), then f is a constant function. e. Property 4 of Table 5.3 implies that \(\int _a^b\ xf(x)\ dx\ =\ x\int _a^b\ f(x)\ dx\).

Approximating definite integrals Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration. b. Calculate \(\triangle x\) and the grid points \(x_0,\ x_1,\ .\ .\ .\ ,\ x_n\), assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. \(\int _0^2\ (x^2-2)\ dx;\ \ n=4\)

Approximating definite integrals Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration. b. Calculate \(\triangle x\) and the grid points \(x_0,\ x_1,\ .\ .\ .\ ,\ x_n\), assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. \(\int _3^6\ (1-2x)\ dx;\ \ n=6\)

Approximating definite integrals Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration. b. Calculate \(\triangle x\) and the grid points \(x_0,\ x_1,\ .\ .\ .\ ,\ x_n\), assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. \(\int _0^{\pi/2} \cos x\ dx;\ \ n=4\)

Approximating definite integrals Complete the following steps for the given integral and the given value of n. a. Sketch the graph of the integrand on the interval of integration. b. Calculate \(\triangle x\) and the grid points \(x_0,\ x_1,\ .\ .\ .\ ,\ x_n\), assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of n. d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. \(\int _1^7\ \frac{1}{x}\ dx;\ \ n=6\)

Approximating definite integrals with a calculator Consider the following definite integrals. a. Write the left and right Riemann sums in sigma notation for n = 20, 50, and 100. Then evaluate the sums using a calculator. b. Based upon your answers to part (a), make a conjecture about the value of the definite integral. \(\int _4^9\ 3\sqrt{x}\ dx\)

Approximating definite integrals with a calculator Consider the following definite integrals. a. Write the left and right Riemann sums in sigma notation for n = 20, 50, and 100. Then evaluate the sums using a calculator. b. Based upon your answers to part (a), make a conjecture about the value of the definite integral. \(\int _0^1\ (x^2+1)\ dx\)

Approximating definite integrals with a calculator Consider the following definite integrals. a. Write the left and right Riemann sums in sigma notation for n = 20, 50, and 100. Then evaluate the sums using a calculator. b. Based upon your answers to part (a), make a conjecture about the value of the definite integral. \(\int _0^1 \tan\ (\frac{\pi x}{4})\ dx\)

Approximating definite integrals with a calculator Consider the following definite integrals. a. Write the left and right Riemann sums in sigma notation for n = 20, 50, and 100. Then evaluate the sums using a calculator. b. Based upon your answers to part (a), make a conjecture about the value of the definite integral. \(\int _0^1\ e^x\ dx\)

Approximating definite integrals with a calculator Consider the following definite integrals. a. Write the left and right Riemann sums in sigma notation for n = 20, 50, and 100. Then evaluate the sums using a calculator. b. Based upon your answers to part (a), make a conjecture about the value of the definite integral. \(\int _{-1}^1 \cos\ (\frac{x\pi}{2})\ dx\)

Riemann sums with midpoints and a calculator Consider the following definite integrals. a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of n. b. Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. \(\int _1^4\ 2\sqrt{x}\ dx\)

Riemann sums with midpoints and a calculator Consider the following definite integrals. a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of n. b. Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. \(\int _{-1}^2 \sin\ (\frac{x \pi}{4})\ dx\)

Riemann sums with midpoints and a calculator Consider the following definite integrals. a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of n. b. Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. \(\int _0^4\ (4x-x^2)\ dx\)

Riemann sums with midpoints and a calculator Consider the following definite integrals. a. Write the midpoint Riemann sum in sigma notation for an arbitrary value of n. b. Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral. \(\int _0^{1/2} \sin^{-1}x\ dx\)

More properties of integrals consider two functions f and g on [1, 6] such that \(\int _1^6\ f(x)\ dx = 10,\ \int _1^6\ g(x)\ dx=5,\ \int _4^6\ f(x)\ dx=5\), and \(\int _1^4\ g(x)\ dx=2\). Evaluate the following integrals. a. \(\int _1^4\ 3f(x)\ dx\) b. \(\int _1^6\ (f(x)-g(x))\ dx\) c. \(\int _1^4\ (f(x)-g(x))\ dx\) d. \(\int _4^6\ (g(x)-f(x))\ dx\) e. \(\int _4^6\ 8g(x)\ dx\) f. \(\int _4^1\ 2f(x)\ dx\)

Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area the net area of the region described. The region between the graph of y = 4x - 8 and the x-axis for \(-4\ \leq\ x\ \leq\ 8\)

Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area the net area of the region described. The region between the graph of y = -3x and x-axis for \(-2\ \leq\ x\ \leq\ 2\)

Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area the net area of the region described. The region between the graph of y = 3x - 6 and the x-axis for \(0\ \leq\ x\ \leq\ 6\)

Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area the net area of the region described. The region between the graph of y = 1 - |x| and the x-axis for \(-2\ \leq\ x\ \leq\ 2\)

Area by geometry Use geometry to evaluate the following integrals. \(\int _{-2}^3\ |x+1|\ dx\)

Area by geometry Use geometry to evaluate the following integrals. \(\int _1^6\ |2x-4|\ dx\)

Area by geometry Use geometry to evaluate the following integrals. \(\int _1^6\ (3x-6)\ dx\)

Area by geometry Use geometry to evaluate the following integrals. \(\int _{-6}^4\ \sqrt{24-2x-x^2}\ dx\)

Integrating piecewise continuous functions Suppose f is continuous on the interval [a, c] and on the interval (c, b], where a < c < b, with a finite jump at x = c. Form a uniform partition on the interval [a, c] with n grid points and another uniform partition on the interval [c, b] with m grid points, where x = c is a grid point of both partitions. Write a Riemann sum for \(\int _a^b\ f(x)\ dx\) and separate it into two pieces for [a, c] and [c, b]. Explain why \(\int _a^b\ f(x)\ dx\ =\ \int _a^c\ f(x)\ dx\ +\ \int _c^b\ f(x)\ dx\).

Piecewise continuous functions Use geometry and the result of Exercise 74 to evaluate the following integrals. \(\int _0^{10}\ f(x)\ dx\) where \(f(x)=\left\{\begin{array}{ll} 2 & \text { if } 0 \leq x<5 \\ 3 & \text { if } 5 \leq x \leq 10 \end{array}\right.\)

Piecewise continuous functions Use geometry and the result of Exercise 74 to evaluate the following integrals. \(\int _1^6\ f(x)\ dx\) where \(f(x)=\left\{\begin{array}{ll} 2 x & \text { if } 1 \leq x<4 \\ 10-2 x & \text { if } 4 \leq x \leq 6 \end{array}\right.\)

Constants in integrals Use the definition of the definite integral to justify the property \(\int _a^b\ cf(x)\ dx\ =\ c\int _a^b\ f(x)\ dx\), where f is continuous and c is a real number.

Exact area Consider the linear function f(x) = 2px + q on the interval [a, b], where a, b, p, and q are positive constants. a. Show that the midpoint Riemann sum with n subintervals equals \(\int _a^b\ f(x)\ dx\) for any value of n. b. Show that \(\int _a^b\ f(x)\ dx=(b-a)[p(b+a)+q]\).

A non-integrable function Consider the function defined on [0, 1] such that f(x) = 1 if x is a rational number and f(x) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral \(\int _0^1\ f(x)\ dx\) does not exist. Show that if you consider only right, left, and midpoint Riemann sums on regular partitions with n subintervals, then they equal 1 for all n.

Powers of x by Riemann sums Consider the integral \(I(p)=\int _0^1\ x^p\ dx\) where p is a positive integer. a. Write the left Riemann sum for the integral with n subintervals. b. It is a fact (proved by the 17th-century mathematicians Fermat and Pascal) that \(\lim _{n \rightarrow \infty}\ \frac{1}{n}\ \sum _{k=0}^{n-1}\ (\frac{k}{n})^p\ =\ \frac{1}{p+1}\) Use this fact to evaluate I(p).