In Exercises 1922, calculate the Riemann sum R(f, P , C)for the given function

What is  5 3 dx [the function is f (x) = 1]?

Let I =  7 2 f (x) dx, where f is continuous. State whether the following are true or false: (a) I is the area between the graph and the x-axis over [2, 7]. (b) If f (x) 0, then I is the area between the graph and the x-axis over [2, 7]. (c) If f (x) 0, then I is the area between the graph of f and the x-axis over [2, 7].

Explain graphically:  0 cos x dx = 0

Which is negative,  5 1 8 dx or  1 5 8 dx?

In Exercises 110, draw a graph of the signed area represented by the integral and compute it using geometry 8 6 (7 x) dx

In Exercises 110, draw a graph of the signed area represented by the integral and compute it using geometry 3/2 /2 sin x dx

In Exercises 110, draw a graph of the signed area represented by the integral and compute it using geometry 5 0  25 x2 dx

In Exercises 110, draw a graph of the signed area represented by the integral and compute it using geometry 3 2 |x| dx

In Exercises 110, draw a graph of the signed area represented by the integral and compute it using geometry 2 2 (2 |x|)dx

In Exercises 110, draw a graph of the signed area represented by the integral and compute it using geometry 5 2 (3 + x 2|x|)dx

Calculate  10 0 (8 x) dx in two ways: (a) As the limit lim N RN (b) By sketching the relevant signed area and using geometry

Calculate  4 1 (4x 8)dx in two ways: (a) As the limit lim N RN (b) By using geometry

In Exercises 13 and 14, refer to Figure 14. 13. Evaluate: (a)  2 0 f (x) dx (b)  6 0 f (x) dx

Evaluate: (a)  4 1 f (x) dx (b)  6 1 |f (x)| dx y = f(x)

In Exercises 15 and 16, refer to Figure 15. 15. Evaluate  3 0 g(t) dt and  5 3 g(t) dt.

Find a, b, and c such that  a 0 g(t) dt and  c b g(t) dt are as large as possible. 12345 2 1 1 2 y = g(t) t y

Describe the partition P and the set of sample points C for the Riemann sum shown in Figure 16. Compute the value of the Riemann sum. x 0.5 4.5 5 1 3 2 2.5 3.2 34.25 20 15 8 y

Compute R(f, P , C) for f (x) = x2 + x for the partition P and the set of sample points C in Figure 16. [The curve shown is not f (x) = x2 + x.]

In Exercises 1922, calculate the Riemann sum R(f, P , C)for the given function, partition, and choice of sample points. Also, sketch the graph of f and the rectangles corresponding to R(f, P , C). 19. f (x) = x, P = {1, 1.2, 1.5, 2}, C = {1.1, 1.4, 1.9}

f (x) = 2x + 3, P = {4, 1, 1, 4, 8}, C = {3, 0, 2, 5}

f (x) = x2 + x, P = {2, 3, 4.5, 5}, C = {2, 3.5, 5}

f (x) = sin x, P =  0, 6 , 3 , 2  , C = {0.4, 0.7, 1.2}

In Exercises 2328, sketch the signed area represented by the integral. Indicate the regions of positive and negative area. 23.  5 0 (4x x2)dx

In Exercises 2328, sketch the signed area represented by the integral. Indicate the regions of positive and negative area./4 /4 tan x dx

In Exercises 2328, sketch the signed area represented by the integral. Indicate the regions of positive and negative area.2 sin x dx

In Exercises 2328, sketch the signed area represented by the integral. Indicate the regions of positive and negative area.3 0 sin x dx

In Exercises 2328, sketch the signed area represented by the integral. Indicate the regions of positive and negative area.2 1/2 ln x dx

In Exercises 2328, sketch the signed area represented by the integral. Indicate the regions of positive and negative area.1 1 tan1 x dx

In Exercises 2932, determine the sign of the integral without calculating it. Draw a graph if necessary. 29.  1 2 x4 dx

In Exercises 2932, determine the sign of the integral without calculating it. Draw a graph if necessary.2 x3 dx

In Exercises 2932, determine the sign of the integral without calculating it. Draw a graph if necessary. 2 0 x sin x dx

In Exercises 2932, determine the sign of the integral without calculating it. Draw a graph if necessary.2 0 sin x x dx

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals. 33.  4 0 (6t 3)dt

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.2 3 (4x + 7)dx

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.9 0 x2 dx

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.5 2 x2 dx

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.1 0 (u2 2u) du

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.1/2 0 (12y2 + 6y) dy

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.1 3 (7t 2 + t + 1)dt

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.3 3 (9x 4x2)dx

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.1 a (x2 + x) dx

In Exercises 3342, use properties of the integral and the formulas in the summary to calculate the integrals.a2 a x2 dx

In Exercises 4347, calculate the integral, assuming that  5 0 f (x) dx = 5,  5 0 g(x) dx = 12 43.  5 0 (f (x) + g(x)) dx

In Exercises 4347, calculate the integral, assuming that  5 0 f (x) dx = 5,  5 0 g(x) dx = 12 5 0  2f (x) 1 3 g(x) dx

In Exercises 4347, calculate the integral, assuming that  5 0 f (x) dx = 5,  5 0 g(x) dx = 12 0 5 g(x) dx

In Exercises 4347, calculate the integral, assuming that  5 0 f (x) dx = 5,  5 0 g(x) dx = 12 5 0 (f (x) x) dx

Is it possible to calculate  5 0 g(x)f (x) dx from the information given?

Prove by computing the limit of right-endpoint approximations:  b 0 x3 dx = b4 4

In Exercises 4954, evaluate the integral using the formulas in the summary and Eq. (9). 49.  3 0 x3 dx

In Exercises 4954, evaluate the integral using the formulas in the summary and Eq. (9) 3 1 x3 dx

In Exercises 4954, evaluate the integral using the formulas in the summary and Eq. (9) 3 0 (x x3)dx

In Exercises 4954, evaluate the integral using the formulas in the summary and Eq. (9) 1 0 (2x3 x + 4)dx

In Exercises 4954, evaluate the integral using the formulas in the summary and Eq. (9) 1 0 (12x3 + 24x2 8x) dx

In Exercises 4954, evaluate the integral using the formulas in the summary and Eq. (9) 2 2 (2x3 3x2)dx

In Exercises 5558, calculate the integral, assuming that  1 0 f (x) dx = 1,  2 0 f (x) dx = 4,  4 1 f (x) dx = 7 55.  4 0 f (x) dx

In Exercises 5558, calculate the integral, assuming that  1 0 f (x) dx = 1,  2 0 f (x) dx = 4,  4 1 f (x) dx = 7 2 1 f (x) dx

In Exercises 5558, calculate the integral, assuming that  1 0 f (x) dx = 1,  2 0 f (x) dx = 4,  4 1 f (x) dx = 7 1 4 f (x) dx

In Exercises 5558, calculate the integral, assuming that  1 0 f (x) dx = 1,  2 0 f (x) dx = 4,  4 1 f (x) dx = 7 4 2 f (x) dx

In Exercises 5962, express each integral as a single integral. 59.  3 0 f (x) dx +  7 3 f (x) dx

In Exercises 5962, express each integral as a single integral.9 2 f (x) dx  9 4 f (x) dx

In Exercises 5962, express each integral as a single integral.9 2 f (x) dx  5 2 f (x) dx

In Exercises 5962, express each integral as a single integral.3 7 f (x) dx +  9 3 f (x) dx

In Exercises 6366, calculate the integral, assuming that f is integrable and  b 1 f (x) dx = 1 b1 for all b > 0. 63.  5 1 f (x) dx

In Exercises 6366, calculate the integral, assuming that f is integrable and  b 1 f (x) dx = 1 b1 for all b > 0. 5 3 f (x) dx

In Exercises 6366, calculate the integral, assuming that f is integrable and  b 1 f (x) dx = 1 b1 for all b > 0.6 1 (3f (x) 4)dx

In Exercises 6366, calculate the integral, assuming that f is integrable and  b 1 f (x) dx = 1 b1 for all b > 0.1 1/2 f (x) dx

Explain the difference in graphical interpretation between  b a f (x) dx and  b a |f (x)| dx.

Use the graphical interpretation of the definite integral to explain the inequality       b a f (x) dx       b a |f (x)| dx where f is continuous. Explain also why equality holds if and only if either f (x) 0 for all x or f (x) 0 for all x.

Let f (x) = x. Find an interval [a, b] such that       b a f (x) dx      = 1 2 and  b a |f (x)| dx = 3 2

Evaluate I =  2 0 sin2 x dx and J =  2 0 cos2 x dx as follows. First show with a graph that I = J . Then prove that I + J = 2.

In Exercises 7174, calculate the integral. 71.  6 0 |3 x| dx

In Exercises 7174, calculate the integral.3 1 |2x 4| dx

In Exercises 7174, calculate the integral.1 1 |x3| dx

In Exercises 7174, calculate the integral.2 0 |x2 1| dx

Use the Comparison Theorem to show that  1 0 x5 dx  1 0 x4 dx,  2 1 x4 dx  2 1 x5 dx

Prove that 1 3  6 4 1 x dx 1 2 .

Prove that 0.0198  0.3 0.2 sin x dx 0.0296. Hint: Show that 0.198 sin x 0.296 for x in [0.2, 0.3]

Prove that 0.277  /4 /8 cos x dx 0.363.

Prove that 0  /2 /4 sin x x dx 2 2 .

Find upper and lower bounds for  1 0 dx  5x3 + 4 .

Suppose that f (x) g(x) on [a, b]. By the Comparison Theorem,  b a f (x) dx  b a g(x) dx. Is it also true that f (x) g (x) for x [a, b]? If not, give a counterexample

State whether true or false. If false, sketch the graph of a counterexample. (a) If f (x) > 0, then  b a f (x) dx > 0. (b) If  b a f (x) dx > 0, then f (x) > 0.

Explain graphically: If f is an odd function, then  a a f (x) dx = 0.

Compute  1 1 (sin x)(sin2 x + 1)dx.

Let k and b be positive. Show, by comparing the right-endpoint approximations, that  b 0 xk dx = bk+1  1 0 xk dx

Verify for 0 b 1 by interpreting in terms of area:  b 0  1 x2 dx = 1 2 b  1 b2 + 1 2 sin1 b

Suppose that f and g are continuous functions such that, for all a,  a a f (x) dx =  a a g(x) dx Give an intuitive argument showing that f (0) = g(0). Explain your idea with a graph.

Theorem 4 remains true without the assumption a b c. Verify this for the cases b